8 Rigid subsets of symplectic manifolds
Michael Entova and Leonid Polterovichb
August 31, 2018
Abstract
We show that there is an hierarchy of intersection rigidity proper-
ties of sets in a closed symplectic manifold: some sets cannot be dis-
placed by symplectomorphisms from more sets than the others. We
also find new examples of rigidity of intersections involving, in par-
ticular, specific fibers of moment maps of Hamiltonian torus actions,
monotone Lagrangian submanifolds (following the works of P.Albers
and P.Biran-O.Cornea), as well as certain, possibly singular, sets de-
fined in terms of Poisson-commutative subalgebras of smooth func-
tions. In addition, we get some geometric obstructions to semi-simpli-
city of the quantum homology of symplectic manifolds. The proofs are
based on the Floer-theoretical machinery of partial symplectic quasi-
states.
aPartially supported by E. and J. Bishop Research Fund and by the Israel Science
Foundation grant # 881/06.
bPartially supported by the Israel Science Foundation grant # 11/03.
http://arxiv.org/abs/0704.0105v2
Contents
1 Introduction and main results 3
1.1 Many facets of displaceability . . . . . . . . . . . . . . . . . . 3
1.2 Preliminaries on quantum homology . . . . . . . . . . . . . . . 8
1.3 An hierarchy of rigid subsets within Floer theory . . . . . . . 10
1.4 Hamiltonian torus actions . . . . . . . . . . . . . . . . . . . . 14
1.5 Super(heavy) monotone Lagrangian submanifolds . . . . . . . 19
1.6 An effect of semi-simplicity . . . . . . . . . . . . . . . . . . . . 23
1.7 Discussion and open questions . . . . . . . . . . . . . . . . . . 27
1.7.1 Strong displaceability beyond Floer theory? . . . . . . 27
1.7.2 Heavy fibers of Poisson-commutative subspaces . . . . 28
2 Detecting stable displaceability 32
3 Preliminaries on Hamiltonian Floer theory 33
3.1 Valuation on QH∗(M) . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Hamiltonian Floer theory . . . . . . . . . . . . . . . . . . . . 34
3.3 Conley-Zehnder and Maslov indices . . . . . . . . . . . . . . . 36
3.4 Spectral numbers . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Partial symplectic quasi-states . . . . . . . . . . . . . . . . . . 44
4 Basic properties of (super)heavy sets 45
5 Products of (super)heavy sets 48
5.1 Product formula for spectral invariants . . . . . . . . . . . . . 48
5.2 Decorated Z2-graded complexes . . . . . . . . . . . . . . . . . 49
5.3 Reduced Floer and Quantum homology . . . . . . . . . . . . . 50
5.4 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Proof of algebraic Theorem 5.2 . . . . . . . . . . . . . . . . . 52
6 Stable non-displaceability of heavy sets 57
7 Analyzing stable stems 59
8 Monotone Lagrangian submanifolds 61
9 Rigidity of special fibers of Hamiltonian actions 66
9.1 Calabi and mixed action-Maslov . . . . . . . . . . . . . . . . . 76
1 Introduction and main results
1.1 Many facets of displaceability
A well-studied and easy to visualize rigidity property of subsets of a symplec-
tic manifold (M,ω) is the rigidity of intersections: a subset X ⊂ M cannot
be displaced from the closure of a subset Y ⊂ M by a compactly supported
Hamiltonian isotopy:
φ(X) ∩ Y 6= ∅ ∀φ ∈ Ham(M) .
We say in such a case that X cannot be displaced from Y . If X cannot be
displaced from itself we call it non-displaceable. These properties become
especially interesting and purely symplectic when X can be displaced from
itself or from Y by a (compactly supported) smooth isotopy.
One of the main themes of the present paper is that “some non-displace-
able sets are more rigid than others.” To explain this, we need the following
ramifications of the notion of a non-displaceable set:
Strong non-displaceability: A subset X ⊂ M is called strongly non-
displaceable if one cannot displace it by any (not necessarily Hamiltonian)
symplectomorphism of (M,ω).
Stable non-displaceability: Consider T ∗S1 = R × S1 with the coordi-
nates (r, θ) and the symplectic form dr ∧ dθ. We say that X ⊂ M is stably
non-displaceable if X × {r = 0} is non-displaceable in M × T ∗S1 equipped
with the split symplectic form ω̄ = ω ⊕ (dr ∧ dθ). Let us mention that de-
tecting stably non-displaceable subsets is useful for studying geometry and
dynamics of Hamiltonian flows (see for instance [50] for their role in Hofer’s
geometry and [51] for their appearance in the context of kick stability in
Hamiltonian dynamics).
Formally speaking, the properties of strong and stable non-displaceability
are mutually independent and both are strictly stronger than displaceability.
In the present paper we refine the machinery of partial symplectic quasi-
states introduced in [23] and get new examples of stably non-displaceable
sets, including certain fibers of moment maps of Hamiltonian torus actions
as well as monotone Lagrangian submanifolds discussed by Albers [2] and
Biran-Cornea [15]. Further, we address the following question: given the class
of stably non-displaceable sets, can one distinguish those of them which are
also strongly non-displaceable by means of the Floer theory? Or, other way
around, what are the Floer-homological features of stably non-displaceable
but strongly displaceable sets? Toy examples are given by the equator of the
symplectic two-sphere and by the meridian on a symplectic two-torus. Both
are stably non-displaceable since their Lagrangian Floer homologies are non-
trivial. On the other hand, the equator is strongly non-displaceable, while
the meridian is strongly displaceable by a non-Hamiltonian shift. Later on we
shall explain the difference between these two examples from the viewpoint
of Hamiltonian Floer homology and present various generalizations.
The question on Floer-homological characterization of (strongly) non-displa-
ceable but stably displaceable sets is totally open, see Section 1.7.1 below for
an example involving Gromov’s packing theorem and discussion.
Leaving Floer-theoretical considerations for the next section, let us outline
(in parts, informally) the general scheme of our results: Given a symplectic
manifold (M,ω), we shall define (in the language of the Floer theory) two
collections of closed subsets of M , heavy subsets and superheavy subsets.
Every superheavy subset is heavy, but, in general, not vice versa. Formally
speaking, the hierarchy heavy-superheavy depends in a delicate way on the
choice of an idempotent in the quantum homology ring ofM . This and other
nuances will be ignored in this outline. The key properties of these collections
are as follows (see Theorems 1.2 and 1.5 below):
Invariance: Both collections are invariant under the group of all symplec-
tomorphisms of M .
Stable non-displaceability: Every heavy subset is stably non-displace-
able.
Intersections: Every superheavy subset intersects every heavy subset. In
particular, superheavy subsets are strongly non-displaceable. In contrast to
this, heavy subsets can be mutually disjoint and strongly displaceable.
Products: Product of any two (super)heavy subsets is (super)heavy.
What is inside the collections? The collections of heavy and superheavy
sets include the following examples:
Stable stems: Let A ⊂ C∞(M) be a finite-dimensional Poisson-commuta-
tive subspace (i.e. any two functions from A commute with respect to the
Poisson brackets). Let Φ :M → A∗ be the moment map: 〈Φ(x), F 〉 = F (x).
A non-empty fiber Φ−1(p), p ∈ A∗, is called a stem of A (see [23]) if all
non-empty fibers Φ−1(q) with q 6= p are displaceable and a stable stem if
they are stably displaceable. If a subset of M is a (stable) stem of a finite-
dimensional Poisson-commutative subspace of C∞(M), it will be called just
a (stable) stem. Clearly, any stem is a stable stem. The collection of
superheavy subsets includes all stable stems (see Theorem 1.6 below).
One readily shows that a direct product of stable stems is a stable stem and
that the image of a stable stem under any symplectomorphism is again a
stable stem.
The following example of a stable stem is borrowed (with a minor mod-
ification) from [23]: Let X ⊂ M be a closed subset whose complement is
a finite disjoint union of stably displaceable sets. Then X is a stable stem.
For instance, the codimension-1 skeleton of a sufficiently fine triangulation of
any closed symplectic manifold is a stable stem. Another example is given by
the equator of S2: it divides the sphere into two displaceable open discs and
hence is a stable stem. By taking products, one can get more sophisticated
examples of stable stems. Already the product of equators of the two-spheres
gives rise to a Lagrangian Clifford torus in S2× . . .×S2. To prove its rigidity
properties (such as stable non-displaceability) one has to use non-trivial sym-
plectic tools such as Lagrangian Floer homology, see e.g. [44]. Products of
the 1-skeletons of fine triangulations of the two-spheres can be considered as
singular Lagrangian submanifolds, an object which is currently out of reach
of the Lagrangian Floer theory.
Another example of stable stems comes from Hamiltonian torus actions.
Consider an effective Hamiltonian action ϕ : Tk → Ham(M) with the mo-
ment map Φ = (Φ1, . . . ,Φk) : M → R
k. Assume that Φi is a normalized
Hamiltonian, that is
Φi = 0 for all i = 1, . . . , k. A torus action is called
compressible if the image of the homomorphism ϕ♯ : π1(T
k) → π1(Ham(M)),
induced by the action ϕ, is a finite group. One can show that for compressible
actions the fiber Φ−1(0) is a stable stem (see Theorem 1.7 below).
Special fibers of Hamiltonian torus actions: Consider an effective
Hamiltonian torus action ϕ on a spherically monotone symplectic manifold.
Let I : π1(Ham(M)) → R be the mixed action-Maslov homomorphism intro-
duced in [49]. Since the target space Rk of the moment map Φ is naturally
identified with Hom(π1(T
k),R), the pull back pspec := −ϕ
♯I of the mixed
action-Maslov homomorphism with the reversed sign can be considered as a
point of Rk. The preimage Φ−1(pspec) is called the special fiber of the action.
We shall see below that the special fiber is always non-empty. For monotone
symplectic toric manifolds (that is when 2k = dimM) the special fiber is a
monotone Lagrangian torus. Note that when the action is compressible we
have pspec = 0 and therefore the special fiber is a stable stem according to the
previous example. It is unknown whether the latter property persists for gen-
eral non-compressible actions. Thus in what follows we treat stable stems
and special fibers as separate examples. The collection of superheavy
subsets includes all special fibers (see Theorem 1.9 below).
For instance, consider CP 2 and the Lagrangian Clifford torus in it (i.e.
the torus {[z0 : z1 : z2] ∈ CP
2 | |z0| = |z1| = |z2|}). Take the standard
Hamiltonian T2-action on CP 2 preserving the Clifford torus. It has three
global fixed points away from the Clifford torus. Make an equivariant sym-
plectic blow-up, M , of CP 2 at k of these fixed points, 0 ≤ k ≤ 3, so that
the obtained symplectic manifold is spherically monotone. The torus action
lifts to a Hamiltonian action on M . One can show that its special fiber is
the proper transform of the Clifford torus.
Monotone Lagrangian submanifolds: Let (M2n, ω) be a spherically
monotone symplectic manifold, and let L ⊂ M be a closed monotone La-
grangian submanifold with the minimal Maslov number NL ≥ 2. We say
that L satisfies the Albers condition [2] if the image of the natural morphism
H∗(L;Z2) → H∗(M ;Z2) contains a non-zero element S with
deg S > dimL+ 1−NL .
The collection of heavy sets includes all closed monotone Lagran-
gian submanifolds satisfying the Albers condition (see Theorem 1.15
below).
Specific examples include the meridian on T2, RP n ⊂ CP n and all La-
grangian spheres in complex projective hypersurfaces of degree d in CP n+1
with n > 2d − 3. In the case when the fundamental class [L] of L divides
a non-trivial idempotent in the quantum homology algebra of M , L is, in
fact, superheavy (see Theorem 1.18 below). For instance, this is the case
for RP n ⊂ CP n. Furthermore, a version of superheaviness holds for any
Lagrangian sphere in the complex quadric of even (complex) dimension.
However, there exist examples of heavy, but not superheavy, Lagrangian
submanifolds: For instance, the meridian of the 2-torus is strongly displa-
ceable by a (non-Hamiltonian!) shift and hence is not superheavy. Another
example of heavy but not superheavy Lagrangian submanifold is the sphere
arising as the real part of the Fermat hypersurface
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
with even d ≥ 4 and n > 2d− 3. We refer to Section 1.5 for more details on
(super)heavy monotone Lagrangian submanifolds.
Motivation: Our motivation for the selection of examples appearing in the
list above is as follows. Stable stems provide a playground for studying
symplectic rigidity of singular subsets. In particular, no visible analogue of
the conventional Lagrangian Floer homology technique is applicable to them.
Detecting (stable) non-displaceability of Lagrangian submanifolds via La-
grangian Floer homology is one of the central themes of symplectic topology.
In contrast to this, detecting strong non-displaceabilty has at the moment the
status of art rather than science. That’s why we were intrigued by Albers’
observation that monotone Lagrangian submanifolds satisfying his condition
are in some situations strongly non-displaceable. In the present work we tried
to digest Albers’ results [2] and look at them from the viewpoint of theory
of partial symplectic quasi-states developed in [23]. In addition, our result
on superheaviness of the Lagrangian anti-diagonal in S2 × S2 allows us to
detect an “exotic” monotone Lagrangian torus in this symplectic manifold:
this torus does not intersect the anti-diagonal, and hence is not heavy in
contrast to the standard Clifford torus, see Example 1.20 below.
In [23] we proved a theorem which roughly speaking states that every
(singular) coisotropic foliation has at least one non-displaceable fiber. How-
ever, our proof is non-constructive and does not tell us which specific fibers
are non-displaceable. The notion of the special fiber arose as an attempt to
solve this problem for Hamiltonian circle actions.
Let us mention also that the product property enables us to produce even
more examples of (super)heavy subsets by taking products of the subsets
appearing in the list.
A few comments on the methods involved into our study of heavy and su-
perheavy subsets are in order. These collections are defined in terms of
partial symplectic quasi-states which were introduced in [23]. These are cer-
tain real-valued functionals on C∞(M) with rich algebraic properties which
are constructed by means of the Hamiltonian Floer theory and which conve-
niently encode a part of the information contained in this theory. In general,
the definition of a partial symplectic quasi-state involves the choice of an
idempotent element in the commutative part QH•(M) of the quantum ho-
mology algebra of M . Though the default choice is just the unity of the
algebra, there exist some other meaningful choices, in particular in the case
when QH•(M) is semi-simple. This gives rise to another theme discussed in
this paper: “visible” topological obstructions to semi-simplicity (see Corol-
lary 1.24 and Theorem 1.25 below). For instance, we shall show that if a
monotone symplectic manifold M contains “too many” disjoint monotone
Lagrangian spheres whose minimal Maslov numbers exceed n+ 1, the quan-
tum homology QH•(M) cannot be semi-simple.
Let us pass to the precise set-up. For the reader’s convenience, the ma-
terial presented in this brief outline will be repeated in parts in the next
sections in a less compressed form.
1.2 Preliminaries on quantum homology
The Novikov Ring: Let F denote a base field which in our case will be
either C or Z2, and let Γ ⊂ R be a countable subgroup (with respect to the
addition). Let s, q be formal variables. Define a field KΓ whose elements are
generalized Laurent series in s of the following form:
KΓ :=
θ, zθ ∈ F , ♯
θ > c | zθ 6= 0
<∞, ∀c ∈ R
Define a ring ΛΓ := KΓ[q, q
−1] as the ring of polynomials in q, q−1 with
coefficients in KΓ. We turn ΛΓ into a graded ring by setting the degree of s
to be 0 and the degree of q to be 2.
The ring ΛΓ serves as an abstract model of the Novikov ring associated to
a symplectic manifold. Let (M,ω) be a closed connected symplectic manifold.
Denote by HS2 (M) the subgroup of spherical homology classes in the integral
homology group H2(M ;Z). Abusing the notation we will write ω(A), c1(A)
for the results of evaluation of the cohomology classes [ω] and c1(M) on
A ∈ H2(M ;Z). Set
π̄2(M) := H
2 (M)/ ∼,
where by definition
A ∼ B iff ω(A) = ω(B) and c1(A) = c1(B).
Denote by Γ(M,ω) := [ω](HS2 (M)) ⊂ R the subgroup of periods of the
symplectic form on M on spherical homology classes. By definition, the
Novikov ring of a symplectic manifold (M,ω) is ΛΓ(M,ω). In what follows,
when (M,ω) is fixed, we abbreviate and write Γ, K and Λ instead of Γ(M,ω),
KΓ(M,ω) and ΛΓ(M,ω) respectively.
Quantum homology: Set 2n = dimM . The quantum homology QH∗(M)
is defined as follows. First, it is a graded module over Λ given by
QH∗(M) := H∗(M ;F)⊗F Λ,
with the grading defined by the gradings on H∗(M ;F) and Λ:
deg (a⊗ zsθqk) := deg (a) + 2k .
Second, and most important, QH∗(M) is equipped with a quantum prod-
uct: if a ∈ Hk(M ;F), b ∈ Hl(M ;F), their quantum product is a class
a ∗ b ∈ QHk+l−2n(M), defined by
a ∗ b =
A∈π̄2(M)
(a ∗ b)A ⊗ s
−ω(A)q−c1(A),
where (a ∗ b)A ∈ Hk+l−2n+2c1(A)(M) is defined by the requirement
(a ∗ b)A ◦ c = GW
A (a, b, c) ∀c ∈ H∗(M ;F).
Here ◦ stands for the intersection index and GWFA (a, b, c) ∈ F denotes the
Gromov-Witten invariant which, roughly speaking, counts the number of
pseudo-holomorphic spheres inM in the class A that meet cycles representing
a, b, c ∈ H∗(M ;F) (see [55], [56], [41] for the precise definition).
Extending this definition by Λ-linearity to the whole QH∗(M) one gets
a correctly defined graded-commutative associative product operation ∗ on
QH∗(M) which is a deformation of the classical ∩-product in singular ho-
mology [37], [41], [55], [56], [69]. The quantum homology algebra QH∗(M)
is a ring whose unity is the fundamental class [M ] and which is a module
of finite rank over Λ. If a, b ∈ QH∗(M) have graded degrees deg (a), deg (b)
deg (a ∗ b) = deg (a) + deg (b)− 2n. (1)
We will be mostly interested in the commutative part of the quantum
homology ring (which in the case F = Z2 is, of course, the whole quantum
homology ring). For this purpose we introduce the following notation:
We denote by QH•(M) the whole quantum homology QH∗(M) if
F = Z2 and the even-degree part of QH∗(M) if F = C.
In general, given a topological space X, we denote by H•(X ;F) the
whole singular homology group H∗(X ;F) if F = Z2 and the even-
degree part of H∗(X ;F) if F = C.
Thus, in our notation the ring QH•(M) = H•(M ;F)⊗F Λ is always a com-
mutative subring with unity of QH∗(M) and a module of finite rank over Λ.
We will identify Λ with a subring of QH•(M) by λ 7→ [M ]⊗ λ.
1.3 An hierarchy of rigid subsets within Floer theory
Fix a non-zero idempotent a ∈ QH2n(M) (by obvious grading considera-
tions the degree of every idempotent equals 2n). We shall deal with spectral
invariants c(a,H), where H = Ht : M → R, t ∈ R, is a smooth time-
dependent and 1-periodic in time Hamiltonian function on M , or c(a, φH),
where φH is an element of the universal cover H̃am (M) of Ham(M) rep-
resented by an identity-based path given by the time-1 Hamiltonian flow
generated by H . If H is normalized, meaning that
dimM/2 = 0 for all
t, then c(a,H) = c(a, φH). These invariants, which nowadays are standard
objects of the Floer theory, were introduced in [45] (cf. [59] in the aspherical
case; also see [42],[43] for an earlier version of the construction and [22] for a
summary of definitions and results in the monotone case).
Disclaimer: Throughout the paper we tacitly assume that (M,ω) (as well
as (M ×T2, ω̄), when we speak of stable displaceability) belongs to the class
S of closed symplectic manifolds for which the spectral invariants are well
defined and enjoy the standard list of properties (see e.g. [41, Theorem
12.4.4]). For instance, S contains all symplectically aspherical and spherically
monotone manifolds. Furthermore, S contains all symplectic manifolds M2n
for which, on one hand, either c1 = 0 or the minimal Chern number (on
HS2 (M)) is at least n − 1 and, on the other hand, [ω](H
2 (M)) is a discrete
subgroup of R (cf. [64]). The general belief is that the class S includes all
symplectic manifolds.
Define a functional ζ : C∞(M) → R by
ζ(H) := lim
c(a, lH)
It is shown in [23] that the functional ζ has some very special algebraic
properties (see Theorem 3.6) which form the axioms of a partial symplectic
quasi-state introduced in [23]. The next definition is motivated in part by
the work of Albers [2].
Definition 1.1. A closed subset X ⊂ M is called heavy (with respect to ζ
or with respect to a used to define ζ) if
ζ(H) ≥ inf
H ∀H ∈ C∞(M) , (3)
and is called superheavy (with respect to ζ or a) if
ζ(H) ≤ sup
H ∀H ∈ C∞(M) . (4)
The default choice of an idempotent a is the unity [M ] ∈ QH∗(M). In this
case, as we shall see below, the collections of heavy and superheavy sets
satisfy the properties listed in Section 1.1 and include the examples therein.
In view of potential applications (including geometric obstructions to semi-
simplicity of the quantum homology), we shall work, whenever possible, with
general idempotents.
The asymmetry between supX H and infX H is related to the fact that
the spectral numbers satisfy a triangle inequality c(a ∗ b, φFφG) ≤ c(a, φF ) +
c(b, φG), while there may not be a suitable inequality “in the opposite direc-
tion”. In the case when such an “opposite” inequality exists (e.g. when a = b
is an idempotent and ζ defined by it is a genuine symplectic quasi-state – see
Section 1.6 below) the symmetry between supX H and infX H gets restored
and the classes of heavy and superheavy sets coincide.
Let us emphasize that the notion of (super)heaviness depends on the
choice of a coefficient ring for the Floer theory. In this paper the coefficients
for the Floer theory will be either Z2 or C depending on the situation. Unless
otherwise stated, our results on (super)heavy subsets are valid for any choice
the coefficients.
The group Symp (M) of all symplectomorphisms of M acts naturally on
H∗(M ;F) and hence on QH∗(M) = H∗(M ;F) ⊗F Λ. Clearly, the identity
component Symp0(M) of Symp (M) acts trivially on QH∗(M) and hence for
any idempotent a ∈ QH∗(M) the corresponding ζ is Symp0(M)-invariant.
Thus the image of a (super)heavy set under an element of Symp0(M) is again
a (super)heavy set with respect to the same idempotent a. If a is invariant
under the action of the whole Symp (M) (for instance, if a = [M ]) the classes
of heavy and superheavy sets with respect to a are invariant under the action
of the whole Symp (M) in agreement with the invariance property presented
in Section 1.1 above.
Let us mention also that the collections of (super)heavy sets enjoy a
stability property under inclusions: If X, Y , X ⊂ Y , are closed subsets of M
and X is heavy (respectively, superheavy) with respect to an idempotent a
then Y is also heavy (respectively, superheavy) with respect to the same a.
We are ready now to formulate the main results of the present section.
Theorem 1.2. Assume a and ζ are fixed. Then
(i) Every superheavy set is heavy, but, in general, not vice versa.
(ii) Every heavy subset is stably non-displaceable.
(iii) Every superheavy set intersects every heavy set. In particular, a super-
heavy set cannot be displaced by a symplectic (not necessarily Hamil-
tonian) isotopy and if the idempotent a is invariant under the symplec-
tomorphism group of (M,ω) (e.g. if a = [M ]), every superheavy set is
strongly non-displaceable.
The following theorem discusses the relation between heaviness/super-
heaviness properties with respect to different idempotents. In particular, it
shows that [M ] plays a special role among all the other non-zero idempotents
in QH∗(M).
Theorem 1.3. Assume a is a non-zero idempotent in the quantum homology.
(i) Every set that is superheavy with respect to [M ] is also superheavy with
respect to a.
(ii) Every set that is heavy with respect to a is also heavy with respect to
[M ].
(iii) Assume that the idempotent a is a sum of non-zero idempotents
e1, . . . , el and assume that a closed subset X ⊂ M is heavy with re-
spect to a. Then X is heavy with respect to ei for at least one i.
The next proposition shows that, in general, the heaviness of a set does
depend on the choice of an idempotent in the quantum homology.
Proposition 1.4. Consider the torus T2n equipped with the standard sym-
plectic structure ω = dp∧dq. Let M2n = T2n♯CP n be a symplectic blow-up of
T2n at one point (the blow up is performed in a small ball around the point).
Assume that the Lagrangian torus L ⊂ T2n given by q = 0 does not intersect
the ball in T2n, where the blow up was performed.
Then the proper transform of L (identified with L) is a Lagrangian sub-
manifold of M , which is not heavy with respect to some non-zero idempotent
a ∈ QH∗(M) but heavy with respect to [M ]. (Here we work with F = Z2).
Next, consider direct products of (super)heavy sets. We start with the fol-
lowing convention on tensor products. Let Γi, i = 1, 2, be two countable
subgroups of R. Let Ei be a module over KΓi. We put
E1⊗̂KE2 =
E1 ⊗KΓ1 KΓ1+Γ2
⊗KΓ1+Γ2
E2 ⊗KΓ2 KΓ1+Γ2
. (5)
If E1, E2 are also rings we automatically assume that the middle tensor prod-
uct is the tensor product of rings. In simple words, we extend both modules
to KΓ1+Γ2-modules and consider the usual tensor product over KΓ1+Γ2 .
Given two symplectic manifolds, (M1, ω1) and (M2, ω2), note that the
subgroups of periods of the symplectic forms satisfy
Γ(M1 ×M2, ω1 ⊕ ω2) = Γ(M1, ω1) + Γ(M2, ω2) .
Furthermore, due to the Künneth formula for quantum homology (see e.g.
[41, Exercise 11.1.15] for the statement in the monotone case; the general
case in our algebraic setup can be treated similarly) there exists a natural
ring monomorphism linear over KΓ1+Γ2
QH2n1(M1)⊗̂KQH2n2(M2) →֒ QH2n1+2n2(M1 ×M2) ,
We shall fix a pair of idempotents ai ∈ QH∗(Mi), i = 1, 2. The notions
of (super)heaviness in M1,M2 and M1 ×M2 are understood in the sense of
idempotents a1, a2 and a1 ⊗ a2 respectively.
Theorem 1.5. Assume that Xi is a heavy (resp. superheavy) subset of Mi
with respect to some idempotent ai, i = 1, 2. Then the product X1 × X2
is a heavy (resp. superheavy) subset of M with respect to the idempotent
a1 ⊗ a2 ∈ QH•(M1 ×M2).
An important class of superheavy sets is given by stable stems introduced
and illustrated in Section 1.1.
Theorem 1.6. Every stable stem is a superheavy subset with respect to any
non-zero idempotent a ∈ QH∗(M). In particular, it is strongly and stably
non-displaceable.
In the next section we present an example of stable stems coming from Hamil-
tonian torus actions.
1.4 Hamiltonian torus actions
Fibers of the moment maps of Hamiltonian torus actions form an interesting
playground for testing the various notions of displaceability and heaviness
introduced above. Throughout the paper we deal with effective actions only,
that is we assume that the map ϕ : Tk → Ham(M) defining the action
is a monomorphism. Furthermore, we assume that the moment map Φ =
(Φ1, . . . ,Φk) : M → R
k of the action is normalized: Φi is a normalized
Hamiltonian for all i = 1, . . . , k. By the Atiyah-Guillemin-Sternberg theorem
[6], [30], the image ∆ = Φ(M) of Φ is a k-dimensional convex polytope,
called the moment polytope. The subsets Φ−1(p), p ∈ ∆, are called fibers of
the moment map. A torus action is called compressible if the image of the
homomorphism ϕ♯ : π1(T
k) → π1(Ham(M)), induced by the action ϕ, is a
finite group.
Theorem 1.7. Assume that (M,ω) is equipped with a compressible Hamilto-
nian Tk-action with moment map Φ and moment polytope ∆. Let Y ⊂ ∆ be
any closed convex subset which does not contain 0. Then the subset Φ−1(Y )
is stably displaceable. In particular, the fiber Φ−1(0) is a stable stem.
Note that for symplectic toric manifolds, that is when 2k = dimM , the point
0 is the barycenter of the moment polytope with respect to the Lebesgue
measure. This follows from our assumption on the normalization of the
moment map.
Theorems 1.6 and 1.7 imply that the fiber Φ−1(0) of a compressible torus
action is stably non-displaceable, and thus we get the complete description
of stably displaceable fibers for such actions.
In the case when the action is not compressible, the question of the com-
plete description of stably non-displaceable fibers remains open. We make a
partial progress in this direction by presenting at least one such fiber, called
the special fiber, explicitly in the case when (M,ω) is spherically monotone:
[ω]|HS2 (M)
= κ c1(TM)|HS2 (M)
, κ > 0 .
The special fiber can be described via the mixed action-Maslov homomor-
phism introduced in [49]: Let (M2n, ω) be a spherically monotone symplectic
manifold, and let {ft}, t ∈ [0, 1], be any loop of Hamiltonian diffeomorphisms,
with f0 = f1 = 1, generated by a 1-periodic normalized Hamiltonian func-
tion F (x, t). The orbits of any Hamiltonian loop are contractible due to the
standard Floer theory1. Pick any point x ∈ M and any disc u : D2 → M
spanning the orbit γ = {ftx}. Define the action
2 of the orbit by
AF (γ, u) :=
F (γ(t), t)dt−
u∗ω .
Trivialize the symplectic vector bundle u∗(TM) over D2 and denote by
mF (γ, u) the Maslov index of the loop of symplectic matrices corresponding
to {ft∗} with respect to the chosen trivialization. One readily checks that,
in view of the spherical monotonicity, the quantity
I(F ) := −AF (γ, u)−
mF (γ, u)
does not depend on the choice of the point x and the disc u, and is invariant
under homotopies of the Hamiltonian loop {ft}. In fact, I is a well defined
homomorphism from π1(Ham(M)) to R (see [49], [68]).
Assume again that ϕ : Tk → Ham(M,ω) is a Hamiltonian torus ac-
tion. Write ϕ♯ for the induced homomorphism of the fundamental groups.
Since the target space Rk of the moment map Φ is naturally identified with
Hom(π1(T
k),R), the pull back −ϕ∗♯I of the mixed action-Maslov homomor-
phism with the reversed sign can be considered as a point of Rk. We call
it a special point and denote by pspec. The preimage Φ
−1(pspec) is called the
special fiber of the moment map. In the case k = 1, when Φ is a real-valued
function on M , we will call pspec the special value of Φ.
1The Floer theory guarantees the existence of at least one contractible periodic orbit –
this is not obvious a priori if {ft} is not an autonomous flow. Since all the orbits of {ft}
are homotopic, all of them are contractible.
2Note that our action functional and the one in [49] are of opposite signs.
If k = n and M is a symplectic toric manifold, then pspec can be defined
in purely combinatorial terms involving only the polytope ∆. Namely, pick
a vertex x of ∆. Since ∆ in this case is a Delzant polytope [20], there is a
unique (up to a permutation) choice of vectors v1, . . . ,vn which
• originate at x;
• span the n rays containing the edges of ∆ adjacent to x;
• form a basis of Zn over Z.
Proposition 1.8.
pspec = x+ κ
vi. (6)
Proof. The vertices of the moment polytope are in one-to-one correspondence
with the fixed points of the action. Let x ∈ M be the fixed point corre-
sponding to the vertex x = (x1, . . . ,xn). Then the vectors vj = (v
j , . . . , v
j = 1, . . . , n, are simply the weights of the isotropy Tn-action on TxM . Since
the definition of the mixed action-Maslov invariant of a Hamiltonian circle
action does not depend on the choice of a 1-periodic orbit and a disc span-
ning it, let us compute all Ii, l = 1, . . . , n, using the constant periodic orbit
concentrated at the fixed point x and the constant disc u spanning it. Clearly,
AΦi(x, u) = Φi(x) = xi and mΦi(x, u) = 2
vij ∀i = 1, . . . , n,
which readily yields formula (6).
E.Shelukhin pointed out to us that by summing up equations (6) over all the
vertices x(1), . . . ,x(m) ∈ Rn of the moment polytope, one readily gets that
pspec =
Theorem 1.9. Assume M2n is a spherically monotone symplectic manifold
equipped with a Hamiltonian Tk-action. Then the special fiber of the moment
map is superheavy with respect to any (non-zero) idempotent a ∈ QH2n(M).
In particular, it is stably and strongly non-displaceable.
Let us mention that, in particular, the special fiber is non-empty and so
pspec ∈ ∆. Moreover pspec is an interior point of ∆ – otherwise Φ
−1(pspec) is
isotropic of dimension < n and hence displaceable (see e.g. [9]).
Remark 1.10. If dimM = 2dimTk (that is we deal with a symplectic toric
manifold), the special fiber, say L, is a Lagrangian torus. In fact, this torus
is monotone: for every D ∈ π2(M,L) we have
ω = κ ·mL(D) ,
where mL stands for the Maslov class of L. This is an immediate consequence
of the definitions.
Remark 1.11. Note that when M is spherically monotone and the action is
compressible Theorems 1.7 and 1.9 match each other: in this case pspec = 0
and therefore the special fiber is a stable stem by Theorem 1.7. It is unknown
whether this property persists for the special fibers of non-compressible ac-
tions.
Example 1.12. Let M be the monotone symplectic blow up of CP 2 at k
points (0 ≤ k ≤ 3) which is equivariant with respect to the standard T2-
action and which is performed away from the Clifford torus in CP 2. Since
the blow-up is equivariant, M comes equipped with a Hamiltonian T2-action
extending the T2-action on CP 2. The Clifford torus is a fiber of the moment
map of the T2-action on CP 2. Let L ⊂M be the Lagrangian torus which is
the proper transform of the Clifford torus under the blow-up – it is a fiber of
the moment map of the T2-action on M . Using Proposition 1.8 it is easy to
see that L is the special fiber of M . According to Theorem 1.9, it is stably
and strongly non-displaceable. In fact, it is a stem: the displaceability of
all the other fibers was checked for k = 0 in [10], for k = 1 in [23] and for
k = 2, 3 in [40].
We refer to Section 1.7.2 for further discussion of related problems and very
recent advances.
Digression: Calabi vs. action-Maslov. The method used to prove
Theorem 1.9 also allows to prove the following result involving the mixed
action-Maslov homomorphism. Denote by vol (M) the symplectic volume of
M . Consider the function µ : H̃am (M) → R defined by
µ(φH) := −vol (M) lim
c(a, φlH)/l.
In the case when a is the unity in a field that is a direct summand in the
decomposition of the K-algebra QH2n(M,ω), as an algebra, into a direct
sum of subalgebras, µ is a homogeneous quasi-morphism on H̃am (M) called
Calabi quasi-morphism [22],[24],[46]; in the general case it has weaker prop-
erties [23]. With this language the functional ζ (on normalized functions) is
induced (up to a constant factor) by the pull-back of µ to the Lie algebra of
H̃am (M).
Following P.Seidel we described in [22] the restriction of µ (in fact, for any
spherically monotoneM) on π1(Ham(M)) ⊂ H̃am (M) in terms of the Seidel
homomorphism π1(Ham(M)) → QH
∗ (M), where QH
∗ (M) denotes the
group of invertible elements in the ring QH∗(M). Here we give an alternative
description of µ|π1(Ham(M)) in terms of the mixed action-Maslov homomor-
phism I which, in turn, also provides certain information about the Seidel
homomorphism.
Theorem 1.13. Assume M is spherically monotone and let µ be defined as
above for some non-zero idempotent a ∈ QH∗(M). Then
µ|π1(Ham(M)) = vol (M) · I.
Note that, in particular, µ|π1(Ham(M)) does not depend on a used to de-
fine µ. The theorem also implies that µ descends to a quasi-morphism on
Ham(M) if and only if I : π1(Ham(M)) → R vanishes identically (since µ
descends to a quasi-morphism on Ham(M) if and only if µ|π1(Ham(M)) ≡ 0 –
see e.g. [22], Prop. 3.4). The proof of the theorem is given in Section 9.1.
Let us mention also that, interestingly enough, the homomorphism I
coincides with the restriction to π1(Ham(M)) of yet another quasi-morphism
on H̃am (M) constructed by P.Py (see [52, 53]).
Digression: Action-Maslov homomorphism and Futaki invari-
ant. This remark grew from an observation pointed out to us by Chris
Woodward – we are grateful to him for that. Assume that our symplectic
manifoldM is complex Kähler (i.e. the symplectic structure onM is induced
by the Kähler one) and Fano (by this we mean here that [ω] = c1). Assume
also that a Hamiltonian S1-action {ft} preserves the Kähler metric and the
complex structure. For instance, if M2n is a symplectic toric manifold it can
be equipped canonically with a complex structure and a Kähler metric invari-
ant under the Tn-action on M , hence under the action of any S1-subgroup
{ft} of T
Let V be the Hamiltonian vector field generating the Hamiltonian flow
{ft}. Since {ft} preserves the complex structure, one can associate to V its
Futaki invariant F(V ) ∈ C [29]. It has been checked by E.Shelukhin [63]
that, up to a universal constant factor, this Futaki invariant is equal to the
value of the mixed action-Maslov homomorphism on the loop {ft}:
F(V ) = const · I({ft}).
Note that if such an M admits a Kähler-Einstein metric then the Futaki
invariant has to vanish [29] – thus if I({ft}) 6= 0 the manifold does not admit
a Kähler-Einstein metric. Moreover, if M2n is toric the opposite is also true:
if the Futaki invariant vanishes for any V generating a subgroup of the torus
Tn acting onM thenM admits a Kähler-Einstein metric – this follows from a
theorem by Wang and Zhu [67], combined with a previous result of Mabuchi
[38]. In terms of the moment polytope, the vanishing of the Futaki invariant,
and accordingly the existence of a Kähler-Einstein metric, on a Kähler Fano
toric manifold means precisely that the special point of the polytope coincides
with the barycenter.
1.5 Super(heavy) monotone Lagrangian submanifolds
Let (M2n, ω) be a closed spherically monotone symplectic manifold with [ω] =
κ · c1(TM) on π2(M), κ > 0. Let L ⊂ M be a closed monotone Lagrangian
submanifold with the minimal Maslov number NL ≥ 2. As usually, we put
NL = +∞ if π2(M,L) = 0. As before, we work with the basic field F which
is either Z2 or C. In the case F = C, we assume that L is relatively spin, that
is L is orientable and the 2nd Stiefel-Whitney class of L is the restriction of
some integral cohomology class of M .
Disclaimer: In the case F = C the results of this section are conditional:
We take for granted that Proposition 8.1 below, which was proved by Biran
and Cornea [15] for homologies with Z2-coefficients, extends to homologies
with C-coefficients. In each of the specific examples below we will explicitly
state which F we are using and whenever we use F = C we assume that L
is relatively spin.
Denote by j the natural morphism j : H•(L;F) → H•(M ;F). We say that
L satisfies the Albers condition [2] if there exists an element S ∈ H•(L;F) so
that j(S) 6= 0 and
deg S > dimL+ 1−NL .
We shall refer to such S as to an Albers element of L.
Example 1.14. Assume [L] ∈ H•(L;F) and j([L]) ∈ H•(M ;F) is non-zero.
This means precisely that [L] is an Albers element of L.
A closed monotone Lagrangian submanifold L which satisfies this con-
dition (and whose minimal Maslov number is greater than 1) will be called
homologically non-trivial in M .
Theorem 1.15. Let L be a closed monotone Lagrangian submanifold satisfy-
ing the Albers condition. Then L is heavy with respect to [M ]. In particular,
any homologically non-trivial Lagrangian submanifold is heavy with respect
to [M ].
Example 1.16. Assume that π2(M,L) = 0. Then the homology class of a
point is an Albers element of L, and hence L is heavy. Note that in this
case heaviness cannot be improved to superheaviness: the meridian on the
two-torus is heavy but not superheavy. Here we took F = Z2.
Example 1.17 (Lagrangian spheres in Fermat hypersurfaces). More exam-
ples of heavy (but not necessarily superheavy) monotone Lagrangian sub-
manifolds can be constructed as follows3.
Let M ⊂ CP n+1 be a smooth complex hypersurface of degree d. The
pull-back of the standard symplectic structure from CP n+1 turns M into a
symplectic manifold (of real dimension 2n). If d ≥ 2, then, as it is explained,
for instance, in [12],M contains a Lagrangian sphere: M can be included into
a family of algebraic hypersurfaces of CP n+1 with quadratic degenerations at
isolated points and the vanishing cycle of such a degeneration can be realized
by a Lagrangian sphere following [5], [21], [60], [61], [62].
Let M ⊂ CP n+1 be a projective hypersurface of degree d, 2 ≤ d < n+ 2.
The minimal Chern number of M equals N := n+2− d > 0. Let Ln ⊂ M2n
be a simply connected Lagrangian submanifold (for instance, a Lagrangian
sphere).
First, consider the case when n is even, L is relatively spin and the Euler
characteristics of L does not vanish (this is the case for a sphere). Then the
3We thank P.Biran for his indispensable help with these examples.
homology class j([L]) ∈ Hn(M ;Z) is non-zero: its self-intersection number
in M up to the sign equals the Euler characteristic. Thus [L] is an Albers
element. (Here we use F = C). In view of Theorem 1.15, L is heavy with
respect to [M ].
Second, suppose that n is of arbitrary parity but n > 2d − 3, and no
restriction on the Euler characteristics of L is assumed anymore. This yields
NL = 2N > n+ 1 and thus L satisfies the Albers condition with the class of
a point P as an Albers element. Thus L is heavy with respect to [M ] – here
we use F = Z2.
Finally, fix n ≥ 3 and an even number d such that 4 ≤ d < n+2. Consider
a Fermat hypersurface of degree d
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
n+1 .
Its real part L := M ∩ RP n+1 lies in the affine chart z0 6= 0 and is given by
the equation
xd1 + . . .+ x
n+1 = 1,
where xj := Re(zj/z0) . Since d is even, L is an n-dimensional sphere. As
it was explained above, L is heavy with respect to [M ] if either n is even
(and F = C) or n > 2d − 3 (and F = Z2). However, in either case L is not
superheavy with respect to [M ]. Indeed, let Σd ≈ Zd be the group of complex
roots of unity. Given a vector α = (α1, . . . , αn) ∈ (Σd)
n+1, denote by fα the
symplectomorphism of M given by
fα(z0 : z1 : . . . : zn+1) = (z0 : α1z1 : . . . : αn+1zn+1) . (7)
If all αj ∈ C\R, then αjx /∈ R whenever x ∈ R\{0}, and thus fα(L)∩L = ∅.
Therefore L is strongly displaceable and the claim follows from the part (iii)
of Theorem 1.2.
The next result gives a user-friendly sufficient condition of superheaviness.
Theorem 1.18. Assume L is homologically non-trivial in M and assume
a ∈ QH2n(M) is a non-zero idempotent divisible by j([L]) in QH•(M), that
is a ∈ j([L]) ∗QH•(M). Then L is superheavy with respect to a.
The homological non-triviality of L in the hypothesis of the theorem means
just that [L] is an Albers element of L (see Example 1.14). In fact, the
theorem can be generalized to the cases when L has other Albers elements –
see Remark 8.3 (ii).
Example 1.19 (Lagrangian spheres in quadrics). Here we work with F = C.
Let M be the real part of the Fermat quadric M = {−z20 +
j=1 z
j = 0}.
Assume that n is even and L is a simply connected Lagrangian submanifold
with non-vanishing Euler characteristic (e.g. a Lagrangian sphere). Under
this assumption, [L] ∈ H•(L) and j([L]) 6= 0, since L has non-vanishing self-
intersection. Denote by p ∈ H∗(M ;F) the class of a point. The quantum
homology ring of M was described by Beauville in [8]. In particular, p ∗ p =
w−2[M ], where w = sκnqn. Thus
a± :=
[M ]± pw
are idempotents. One can show that j([L]) divides a− and hence L is a−-
superheavy. Since a− is invariant under the action of Symp(M), the manifold
L is strongly non-displaceable.
For simplicity, we present the calculation in the case n = 2 – the general
case is absolutely analogous. The 2-dimensional quadric is symplectomorphic
to (S2 × S2, ω ⊕ ω). Denote by A and B the classes of [S2] × [point] and
[point] × [S2] respectively. Since the symplectic form vanishes on j([L]) we
get that j([L]) = l(B − A) with l 6= 0. It is known that A ∗ B = p and
B ∗B = w−1[M ]. Thus j([L]) ∗ 1
wB = a−, that is j([L]) divides a−.
In particular, the Lagrangian anti-diagonal
∆ := {(x, y) ∈ S2 × S2 : x = −y} ,
which is diffeomorphic to the 2-sphere, is superheavy with respect to a−. It is
unknown whether ∆ is super-heavy with respect to a+. Further information
on superheavy Lagrangian submanifolds in the quadrics can be extracted
from [15].
Example 1.20 (A non-heavy monotone Lagrangian torus in S2 × S2). Con-
sider the quadric M = S2 × S2 from the previous example. We will think of
S2 as of the unit sphere in R3 whose symplectic form is the area form divided
by 4π. We will work again with F = C. Interestingly enough, such an M
contains a monotone Lagrangian torus that is not heavy with respect to a−.
Namely, consider a submanifold K given by equations4
K = {(x, y) ∈ S2 × S2 : x1y1 + x2y2 + x3y3 = −
, x3 + y3 = 0} .
4We thank Frol Zapolsky for his help with calculations in this example.
One readily checks that K is a monotone Lagrangian torus with NK = 2
which represents a zero element inH2(M ;F) (both with F = C and F = Z2).
Thus H•(K;F) does not contain any Albers element. Furthermore, K is
disjoint from the Lagrangian anti-diagonal ∆ and hence is not heavy with
respect to a− since, as it was shown above, ∆ is superheavy with respect to
a−. In particular, K is an exotic monotone torus: it is not symplectomorphic
to the Clifford torus which is a stem and hence a−-superheavy. A further
study of exotic tori in products of spheres is currently being carried out by
Y.Chekanov and F.Schlenk.
It is an interesting problem to understand whether K is superheavy with
respect to a+, or at least non-displaceable. Identify M \ {the diagonal} with
the unit co-ball bundle of the 2-sphere. After such an identification ∆ corre-
sponds to the zero section, while K corresponds to a monotone Lagrangian
torus, say K ′. Interestingly enough, the Lagrangian Floer homology of K ′
in T ∗S2 (with F = Z2) does not vanish as was shown by Albers and Frauen-
felder in [3], and thus K is not displaceable in M \ {the diagonal}. Thus
the question on (non)-displaceability of K is related to understanding of the
effect of the compactification of the unit co-ball bundle to S2 × S2.
The proofs of theorems above are based on spectral estimates due to
Albers [2] and Biran-Cornea [15]. Furthermore, the results above admit
various generalizations in the framework of Biran-Cornea theory of quantum
invariants for monotone Lagrangian submanifolds, see [15] and the discussion
in Section 8 below.
1.6 An effect of semi-simplicity
Recall that a commutative (finite-dimensional) algebra Q over a field A is
called semi-simple if it splits into a direct sum of fields as follows: Q =
Q1 ⊕ . . .⊕Qd , where
• each Qi ⊂ Q is a finite-dimensional linear subspace over A;
• each Qi is a field with respect to the induced ring structure;
• the multiplication in Q respects the splitting:
(a1, . . . , ad) · (b1, . . . , bd) = (a1b1, . . . , adbd).
A classical theorem of Wedderburn (see e.g. [66], §96) implies that the semi-
simplicity is equivalent to the absence of nilpotents in the algebra.
Remark 1.21. Assume that the K-algebra QH2n(M,ω) splits, as an algebra,
into a direct sum of two algebras, at least one of which is a field, and let e
be the unity in that field. In particular, this is the case when QH2n(M,ω) =
Q1⊕ . . .⊕Qd is semi-simple and e is the unity in one of the fields Qi. A slight
generalization of the argument in [23, 46] (see [24], the remark on pp. 56-57)
shows that the partial quasi-state ζ(e, ·) associated to e is R-homogeneous
(and not just R+-homogeneous as in the general case).
This immediately yields that every set which is heavy with respect to e is
automatically superheavy with respect to e.
In fact, in this situation ζ is a genuine symplectic quasi-state in the sense of
[23] and, in particular, a topological quasi-state in the sense of Aarnes [1] (see
[23] for details). In [1] Aarnes proved an analogue of the Riesz representation
theorem for topological quasi-states which generalizes the correspondence
between genuine states (that is positive linear functionals on C(M)) and
measures. The object τζ corresponding to a quasi-state ζ is called a quasi-
measure (or a topological measure). With this language in place, the sets that
are (super)heavy with respect to ζ are nothing else but the closed sets of the
full quasi-measure τζ . Any two such sets have to intersect for the following
basic reason: any quasi-measure is finitely additive on disjoint closed subsets
and therefore if two closed subsets of M of the full quasi-measure do not
intersect, the quasi-measure of their union must be greater than the total
quasi-measure of M , which is impossible.
Example 1.22. In this example we again assume that F = Z2. Let M =
CP n be equipped with the Fubini-Study symplectic structure ω, normalized
so that [ω] = c1, and let A ∈ H2n−2(M) be the homology class of the hyper-
plane. One readily verifies the following K-algebra isomorphism
QH2n(M) ∼= K[X ]/〈X
n+1 − u−1〉,
where
K = Z2[[u] = {zku
k + zk−1u
k−1 + . . . , zi ∈ Z2 ∀i}
is the field of Laurent-type series in u := sn+1 with coefficients in Z2 and
X = qA. Since no root of degree 2 or more of u−1 is contained in K, the
polynomial P is irreducible over K for any n (see e.g. [34], Theorem 9.1) and
therefore QH2n(M) is a field. Hence the collections of heavy and superheavy
sets with respect to the fundamental class coincide.
We claim that L := RP n ⊂ CP n is superheavy. The case n = 1 cor-
responds to the equator of the sphere, which is known to be a stable stem.
For n ≥ 2, note that NL = n + 1 and S = [RP
2] is an Albers element of L.
Therefore, L is [M ]-heavy by Theorem 1.15, and hence superheavy.
The next result follows directly from Theorem 1.3 (iii) and Remark 1.21:
Theorem 1.23. Assume that QH2n(M) is semi-simple and splits into a
direct sum of d fields whose unities will be denoted by e1, . . . , ed. Assume that
a closed subset X ⊂M is heavy with respect to a non-zero idempotent a – as
one can easily see, such an idempotent has to be of the form a = ej1+ . . .+ejl
for some 1 ≤ j1 < . . . < jl ≤ d. Then X is superheavy with respect to some
eji, 1 ≤ i ≤ l.
The theorem yields the following geometric characterization of non-semi-
simplicity of QH2n(M). Namely, define the symplectic Torelli group as the
group of all symplectomorphisms of M which induce the identity map on
H•(M ;F). For instance, this group contains Symp0(M). Note that any ele-
ment of the symplectic Torelli group acts trivially on the quantum homology
of M and hence maps sets (super)heavy with respect to an idempotent a to
sets (super)heavy with respect to a.
Now Theorem 1.23 readily implies the following
Corollary 1.24. Assume that (M,ω) contains a closed subset X which is
heavy with respect to a non-zero idempotent and displaceable by a symplec-
tomorphism from the symplectic Torelli group. Then QH2n(M) is not semi-
simple.
The simplest examples are provided by sets of the form X×{a meridian}
in M × T2 with a heavy X .
Another result in the same vein is as follows5. Given a set Y of positive
integers, put βY (M) =
i∈Y βi(M), where βi(M) stands for the i-th Betti
number of M over F .
5In the case F = C, Theorem 1.25 is conditional, see the disclaimer in the previous
section.
Theorem 1.25. Assume that either of the following (not mutually excluding)
conditions holds:
(a) M contains m > βY (M) + 1 pair-wise disjoint closed monotone La-
grangian submanifolds whose minimal Maslov numbers are greater than n+1
and belong to a set Y of positive integers.
(b) M contains pair-wise disjoint homologically non-trivial Lagrangian sub-
manifolds6 whose fundamental classes, viewed as (non-zero) elements of
H•(M ;F), are linearly dependent over F .
(In the case F = C assume that all the Lagrangian submanifolds above are
also relatively spin.)
Then QH2n(M) is not semi-simple.
The proof is given in Section 8.
Example 1.26. For instance, if all the Lagrangian submanifolds from part
(a) of the theorem are simply connected, their minimal Maslov numbers are
equal to 2N , so that the set Y consists of one element: Y = {2N}. Thus
if 2N > n + 1 and QH2n(M) is semi-simple, M cannot contain more than
β2N(M)+ 1 pair-wise disjoint simply-connected Lagrangians (provided all of
them are relatively spin if we work with F = C).
Example 1.27. Set F = C. Fix n ≥ 11 and an even number d such that
6 ≤ d < (n + 3)/2. Consider a Fermat hypersurface of degree d
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
As we already saw in Example 1.17, the manifold L := M ∩ RP n+1 is an
n-dimensional Lagrangian sphere. Consider the images fα(L), where sym-
plectomorphisms fα are defined by (7). Note that, as long as αj/βj 6= ±1
for all j, the Lagrangian spheres fα(L) and fβ(L) are disjoint. Using this
observation, it is easy to find d/2 disjoint Lagrangian spheres in M .
The minimal Chern number N ofM equals n+2−d, and so 2N lies in the
interval [n+2, 2n−4]. In this case β2N(M) = 1 (see e.g. [31]). Since d/2 > 2,
we conclude from the previous example that QH2n(M) is not semi-simple.
This conclusion agrees with the computation of QH∗(M) by Beauville [8].
6See Example 1.14 for the definition. As in that example we again assume that all our
Lagrangian submanifolds are closed, monotone and have minimal Maslov number greater
than 1.
It would be interesting to find examples of symplectic manifolds where the
quantum homology is not known a priori and where the above theorems are
applicable. Let us mention that different obstructions to the semi-simplicity
of QH•(M) coming from Lagrangian submanifolds were recently found by
Biran and Cornea [14].
1.7 Discussion and open questions
1.7.1 Strong displaceability beyond Floer theory?
Clearly, displaceability implies stable displaceability. The converse is not
true, as the next example shows:
Example 1.28. Consider the complex projective space CP n equipped with
the Fubini-Study symplectic form (in our normalization the area of a line
equals 1). Identify CP n with the symplectic cut of the Euclidean ball B(1) ⊂
Cn (that is the boundary of B(1) is collapsed to CP n−1 along the fibers of
the Hopf fibration, see [36]), where B(r) := {π|z|2 ≤ r}. Then B(r) ⊂ CP n
(i) displaceable for r < 1/2;
(ii) strongly non-displaceable but stably displaceable for r ∈ [1/2, n/n+1);
(iii) strongly and stably non-displaceable for r ≥ n/n+ 1.
It is instructive to analyze the techniques involved in the proofs: The strong
non-displaceability result in (ii) is an immediate consequence of Gromov’s
packing-by-two-balls theorem, which is proved via the J-holomorphic variant
of the theorem which states that there exists a J-holomorphic line in CP n
passing through any two points. In the case (iii) the ball B(r) contains the
Clifford torus, which is stably non-displaceable. This follows either from the
fact that the Clifford torus is a stem (see [10]), or from non-vanishing of its
Lagrangian Floer homology [16].
The displaceability of B(r) in (i) follows from the explicit construction
of the two balls packing (see [33]). The stable displaceability in (ii) is a
direct consequence of Theorem 1.7 above: Indeed, consider the standard Tn-
action on CP n. The normalized moment polytope ∆ ⊂ Rn has the form
∆ = ∆stand + w where ∆stand is the standard simplex {ρi ≥ 0,
ρi ≤ 1} in
Rn, where (ρ1, . . . , ρn) denote coordinates in R
n, and w = − 1
(1, . . . , 1).
Note that the ball B(r) equals to Φ−1(∆r) where ∆r := r ·∆stand + w. Note
that ∆r does not contain the origin exactly when r ≤
which yields the
stable displaceability in (ii) above.
A mysterious feature of Example 1.28 is as follows. On the one hand, we
believe in the following general empiric principle: whenever one can establish
the non-displaceability of a subset by means of the Floer homology theory,
one gets for free the stable non-displaceability. On the other hand, we be-
lieve, following a philosophical explanation provided by Biran, that Gromov’s
packing-by-two-balls theorem may be extracted from some “operations” in
Floer homology. Example 1.28 shows that at least one of these beliefs is
wrong. It would be interesting to clarify this issue.
1.7.2 Heavy fibers of Poisson-commutative subspaces
It was shown in [23] that for any finite-dimensional Poisson-commutative
subspace A ⊂ C∞(M) at least one of the fibers of its moment map Φ has to
be non-displaceable.
Question. Is it true that at least one fiber of Φ has to be heavy (with respect
to some non-zero idempotent a ∈ QH∗(M))?
It is easy to construct an example of A whose moment map Φ has no
superheavy fibers: take T2 with the coordinates p, q mod 1 on it and take A
to be the set of all smooth functions depending only on p – the corresponding
Φ defines the fibration of T2 by meridians none of which is superheavy.
Here is another question which concerns fibers of symplectic toric man-
ifolds, i.e. fibers of a moment map Φ of an effective Hamiltonian Tn-action
on (M2n, ω). Assume M is (spherically) monotone. Theorem 1.9 shows that
in such a case the special fiber ofM is superheavy, hence stably and strongly
non-displaceable. In all the examples where it has been checked this turns
out to be the only non-displaceable fiber of M .
Question. Is the special fiber for a monotone symplectic toric M always
a stem? In particular, is it the only non-displaceable fiber of the moment
In the monotone case the special fiber is clearly the only heavy fiber of
the moment map, because it is superheavy and any other heavy fiber would
have had to intersect it. On the other hand, if we consider a Hamiltonian Tk-
action on M2n with k < n there can be more than one non-displaceable fiber
of the moment map – for instance, because of purely topological obstructions:
the simplest Hamiltonian T1-action on CP 2 provides such an example. In
the case of monotone symplectic toric manifolds of dimension bigger than 4
the question above is absolutely open.
After the first draft of this paper appeared, a remarkable progress in this
direction has been achieved in the works by Cho [17] and Fukaya, Oh, Ohta
and Ono [28]: In particular, it turns out that a non-monotone symplectic
toric manifold can have more than one non-displaceable fiber – this happens
already for certain equivariant blowups of CP 2.
Organization of the paper:
In Section 2 we prove Theorem 1.7 which in particular states that the
special fiber of a compressible torus action is a stable stem.
In Section 3 we sum up various preliminaries from Floer theory including
basic properties of spectral invariants and partial symplectic quasi-states. In
addition we spell out a useful property of the Conley-Zehnder index: it is a
quasi-morphism on the universal cover of the symplectic group (see Propo-
sition 3.5). For completeness we extract a proof of this property from [54];
alternatively, one can use the results of [19].
In Section 4 we prove parts (i) and (iii) of Theorem 1.2 and Theorem 1.3
on basic properties of (super)heavy sets.
In Section 5 we prove Theorem 1.5 on products of (super)heavy sets. Our
approach is based on a quite general product formula for spectral invariants
(Theorem 5.1), which is proved by a fairly lengthy algebraic argument.
In Section 6 we prove Theorem 1.2 (ii) on stable non-displaceability of
heavy subsets. The argument involves a “baby version” of the above-men-
tioned product formula.
In Section 7 we prove superheaviness of stable stems.
In Section 8 we bring together the proofs of various results related to
(super)heaviness of monotone Lagrangian submanifolds satisfying the Albers
condition, including Theorems 1.15, 1.18, 1.25 and Proposition 1.4.
In Section 9 we prove Theorem 1.9 on superheaviness of special fibers of
Hamiltonian torus actions on monotone symplectic manifolds. The proof is
quite involved. In fact, two tricks enabled us to shorten our original argu-
ment: First, we use the Fourier transform on the space of rapidly decaying
functions on the Lie coalgebra of the torus in order to reduce the problem
to the case of Hamiltonian circle actions. Second, we systematically use the
quasi-morphism property of the Conley-Zehnder index for asymptotic calcu-
lations with Hamiltonian spectral invariants. Finally, in Section 9.1 we prove
Theorem 1.13.
Figure 1 sums up the hierarchy of the non-displaceability properties dis-
cussed above.
������������������������
MHierarchy of non−displaceability properties of a closed subset of 
Heavy
aidempotent 
Superheavy
idempotent  a
wrt a non−zero wrt a non−zero
(3) (4) (5) (6)
action on a spherically
torus action on a (not
a compressible Hamiltonian
necessarily monotone)
Special fiber of
monotone M
Always true.
True under certain conditions (see below)
? Question (under certain conditions − see below)
Monotone
Lagrangian
submanifold L
(14) (15)
(17) (18)
(21) (22)
(16b)
Product of codimension−1
skeletons of fine
triangulations
Strongly
non−displaceable
a Hamiltonian torus
Non−displaceable
a symplectic isotopy
Non−displaceable by
wrt [M]
Heavy
(16a)
Superheavy
wrt [M]
Stable stem
Stably
non−displaceable
Zero fiber of
Figure 1: Hierarchy of non-displaceability properties
(1),(2),(6),(19) - Trivial.
(3) True if a is invariant under the action of the whole group Symp (M) –
Theorem 1.2, part (iii).
(4), (9) Theorem 1.2, part (iii).
(5) True if the algebra QH2n(M) is semi-simple – see Corollary 1.24.
(7a) True if the algebra QH2n(M) splits, as an algebra, into a direct sum of
two algebras, at least one of which is a field, and a is the unity element in
that field – see Remark 1.21.
(7b), (16b) Theorem 1.2, part (i).
(8) Theorem 1.2, part (ii).
(10) Theorem 1.18 (see the assumptions on L there).
(11) True if the algebra QH2n(M) is semi-simple – see Corollary 1.24.
(12) Theorem 1.3, part (i).
(13) Theorem 1.3, part (ii).
(14) Theorem 1.18 (see the assumptions on L there) with a = [M ] – i.e. j(L)
is invertible in QH•(M).
(15) L satisfies the Albers condition – see Theorem 1.15.
(16a) True if QH2n(M) is a field – see Remark 1.21.
(17) Theorem 1.6.
(18) Theorem 1.9.
(20) Theorem 1.7.
(21) Is the special fiber for a monotone symplectic toric M always a stem?
See Section 1.7.2.
(22) True if M is spherically monotone and the torus action is compressible
– see Remark 1.11.
(23) See [23].
2 Detecting stable displaceability
For detecting stable displaceability of a subset of a symplectic manifold we
shall use the following result (cf. [48, Chapter 6]).
Theorem 2.1. Let X be a closed subset of a closed symplectic manifold
(M,ω). Assume that there exists a contractible loop of Hamiltonian diffeo-
morphisms of (M,ω) generated by a normalized time-periodic Hamiltonian
Ht(x) so that Ht(x) 6= 0 for all t ∈ [0, 1] and x ∈ X. Then X is stably
displaceable.
Proof. Denote by ht the Hamiltonian loop generated by H . Let h
t be its
homotopy to the constant loop: h
t = ht and h
t = 1. Write H
(s)(x, t) for
the corresponding normalized Hamiltonians. Consider the family of diffeo-
morphisms Ψs of M × T
∗S1 given by
Ψs(x, r, θ) = (h
θ x, r −H
(s)(h
θ x, θ), θ) .
One readily checks that Ψs, s ∈ [0, 1], is a Hamiltonian isotopy (not com-
pactly supported). We claim that Ψ1 displaces Y := X × {r = 0}. Indeed,
if Ψ1(x, 0, θ) ∈ Y we have hθx ∈ X and Hθ(hθx) = 0 which contradicts the
assumption of the theorem. This completes the proof.
Proof of Theorem 1.7: Choose a linear functional F : Rk → R with
rational coefficients which is strictly positive on Y . Then for some suffi-
ciently large positive integer N the Hamiltonian H := NΦ∗F generates a
contractible Hamiltonian circle action on M and H is strictly positive on
X := Φ−1(Y ). Thus X is stably displaceable in view of the previous theo-
3 Preliminaries on Hamiltonian Floer theory
3.1 Valuation on QH∗(M)
Define a function ν : K → Γ by
θ) = max{ θ | zθ 6= 0} .
The convention is that ν(0) = −∞. In algebraic terms, exp ν is a non-
Archimedean absolute value on K.
The function ν admits a natural extension to Λ and then to QH∗(M) –
abusing the notation we will denote all of them by ν. Namely, any element
of λ ∈ Λ can be uniquely represented as λ =
θ uθs
θ, where each uθ belongs
to F [q, q−1], and any non-zero a ∈ QH∗(M) can be uniquely represented as
i λibi, 0 6= λi ∈ Λ, 0 6= bi ∈ H∗(M ;F). Define
ν(λ) := max
θ | uθ 6= 0
ν(a) := max
ν(λi).
3.2 Hamiltonian Floer theory
We briefly recall the notation and conventions for the setup of the Hamilto-
nian Floer theory that will be used in the proofs.
Let L be the space of all smooth contractible loops γ : S1 = R/Z → M .
We will view such a γ as a 1-periodic map γ : R → M . Let D2 be the
standard unit disk in R2. Consider a covering L̃ of L whose elements are
equivalence classes of pairs (γ, u), where γ ∈ L, u : D2 → M , u|∂D2 = γ (i.e.
u(e2π
−1t) = γ(t)), is a (piecewise smooth) disk spanning γ in M and the
equivalence relation is defined as follows: (γ1, u1) ∼ (γ2, u2) if and only if γ1 =
γ2 and the 2-sphere u1#(−u2) vanishes in H
2 (M). The equivalence class of
a pair (γ, u) will be denoted by [γ, u]. The group of deck transformations of
the covering L̃ → L can be naturally identified with HS2 (M). An element
A ∈ HS2 (M) acts by the transformation
A([γ, u]) = [γ, u#(−A)]. (8)
Let F :M× [0, 1] → R be a Hamiltonian function (which is time-periodic
as we always assume). Set Ft := F (·, t). We will denote by ft the Hamiltonian
flow generated by F , meaning the flow of the time-dependent Hamiltonian
vector field Xt defined by the formula
ω(·, Xt) = dFt(·) ∀t.
(Note our sign convention!)
Let PF ⊂ L be the set of all contractible 1-periodic orbits of the Hamilto-
nian flow generated by F , i.e. the set of all γ ∈ L such that γ(t) = ft(γ(0)).
Denote by P̃F the full lift of PF to L̃.
Denote by Fix (F ) the set of those fixed points of f that are endpoints of
contractible periodic orbits of the flow:
Fix (F ) := {x ∈M | ∃γ ∈ PF , x = γ(0)}.
We say that F is regular if for any x ∈ Fix (F ) the map dxf : TxM → TxM
does not have eigenvalue 1.
Recall that the action functional is defined on L̃ by the formula:
AF ([γ, u]) =
F (γ(t), t)dt−
Note that
AF (Ay) = AF (y) + ω(A) (9)
for all y ∈ L̃ and A ∈ HS2 (M).
For a regular Hamiltonian F define a vector space C(F ) over F as the
set of all formal sums
λiyi, λi ∈ Λ, yi ∈ P̃F ,
modulo the relations
Ay = s−ω(A)q−c1(A)y,
for all y ∈ P̃F , A ∈ H
2 (M). The grading on Λ together with the Conley-
Zehnder index on elements of P̃F (see Section 3.3) defines a Z-grading on
C(F ). We will denote the i-th graded component by Ci(F ).
Given a loop {Jt}, t ∈ S
1, of ω-compatible almost complex structures,
define a Riemannian metric on L by
(ξ1, ξ2) =
ω(ξ1(t), Jtξ2(t))dt,
where ξ1, ξ2 ∈ TγL. Lift this metric to L̃ and consider the negative gradient
flow of the action functional AF . For a generic choice of the Hamiltonian
F and the loop {Jt} (such a pair (F, J) is called regular) the count of iso-
lated gradient trajectories connecting critical points of AF gives rise in the
standard way [26], [32], [58] to a Morse-type differential
d : C(F ) → C(F ), d2 = 0. (10)
The differential d is Λ-linear and has the graded degree −1. It strictly de-
creases the action. The homology, defined by d, is called the Floer homology
and will be denoted by HF∗(F, J). It is a Λ-module. Different choices of a
regular pair (F, J) lead to natural isomorphisms between the Floer homology
groups.
The following proposition summarizes a few basic algebraic properties of
Floer complexes and Floer homology that will be important for us further.
The proof is straightforward and we omit it.
Proposition 3.1.
1) Each Ci(F ) and each HFi(F, J), i ∈ Z, is a finite-dimensional vector
space over K.
2) Multiplication by q defines isomorphisms Ci(F ) → Ci+2(F ) and
HFi(F, J) → HFi+2(F, J) of K-vector spaces.
3) For each i ∈ Z there exists a basis of Ci(F ) over K consisting of the
elements of the form ql[γ, u], with [γ, u] ∈ P̃F .
4) A finite collection of elements of the form ql[γ, u], [γ, u] ∈ P̃F , lying
in C0(F ) ∪C1(F ) is a basis of the vector space C0(F )⊕C1(F ) over the field
K if and only if it is a basis of the module C(F ) over the ring Λ.
3.3 Conley-Zehnder and Maslov indices
In this section we briefly outline the definition and recall the relevant proper-
ties of the Conley-Zehnder index referring to [54, 58, 57] for details. In par-
ticular, we show that the Conley-Zehnder index is a quasi-morphism on the
universal cover S̃p (2k) of the symplectic group Sp(2k) (see Proposition 3.5
below), a fact which will be useful for asymptotic calculations with Floer
homology in the next sections. There are several routes leading to this fact,
which is quite natural since all homogeneous quasi-morphisms on S̃p (2k) are
proportional, and hence the same quasi-morphism admits quite dissimilar
definitions [7]. We extract the quasi-morphism property from the paper of
Robbin and Salamon [54] by bringing together several statements contained
therein7.
The Conley-Zehnder index assigns to each [γ, u] ∈ P̃F a number. Orig-
inally the Conley-Zehnder index was defined only for regular Hamiltonians
[18] – in this case it is integer-valued and gives rise to a grading of the ho-
mology groups in Floer theory. Later the definition was extended in different
ways by different authors to arbitrary Hamiltonians. We will use such an ex-
tension introduced in [54] (also see [57, 58]). In this case the Conley-Zehnder
index may take also half-integer values.
Let k be a natural number. Consider the symplectic vector space R2k
with a symplectic form ω2k on it. Denote by p = (p1, . . . , pk), q = (q1, . . . , qk)
the corresponding Darboux coordinates on the vector space R2k.
7We thank V.L. Ginzburg for stimulating discussions on the material of this section.
Robbin-Salamon index of Lagrangian paths: Let V ⊂ R2k be a
Lagrangian subspace. Consider the Grassmannian Lagr (k) of all Lagrangian
subspaces in R2k and consider the hypersurface ΣV ⊂ Lagr (k) formed by all
the Lagrangian subspaces that are not transversal to V . To such a V and
to any smooth path {Lt}, 0 ≤ t ≤ 1, in Lagr (k) Robbin and Salamon [54]
associate an index, which may take integer or half-integer values and which
we will denote by RS({Lt}, V ). The definition of the index can be outlined
as follows.
A number t ∈ [0, 1] is called a crossing if Lt ∈ ΣV . To each crossing t one
associates a certain quadratic form Qt on the space L(t) ∩ V – see [54] for
the precise definition. The crossing t is called regular if the quadratic form
Qt is non-degenerate. The index of such a regular crossing t is defined as the
signature of Qt if 0 < t < 1 and as half of the signature of Qt if t = 0, 1.
One can show that regular crossings are isolated. For a path {Lt} with only
regular crossings the index RS({Lt}, V ) is defined as the sum of the indices
of its crossings. An arbitrary path can be perturbed, keeping the endpoints
fixed, into a path with only regular crossings and the index of the perturbed
path does not depend on the perturbation – in fact, it depends only on the
fixed endpoints homotopy class of the path. Moreover, it is additive with
respect to the concatenation of paths and satisfies the naturality property:
RS({ALt}, AV ) = RS({Lt}, V ) for any symplectic matrix A.
Indices of paths in Sp (2k): Consider the group Sp (2k) of symplectic
2k × 2k-matrices. Denote by S̃p (2k) its universal cover. One can use the
index RS in order to define two indices on the space of smooth paths in
Sp (2k).
The first index, denoted by Ind2k, is defined as follows. Fix a Lagrangian
subspace V ⊂ R2k. For each smooth path {At}, 0 ≤ t ≤ 1, in Sp (2k) define
Ind2k ({At}, V ) as
Ind2k ({At}, V ) := RS({AtV }, V ).
The naturality of the RS index implies that
RS({BAtB
−1(BV )}, BV ) = RS({BAtV )}, BV ) =
= RS({AtV )}, V ) for any B ∈ Sp (2k)
and thus we get the following naturality condition for Ind2k:
Ind2k ({BAtB
−1}, BV ) = Ind2k ({At}, V ) for any B ∈ Sp (2k). (11)
The second index, which we will call the Conley-Zehnder index of a matrix
path and which will be denoted by CZmatr, is defined as follows. For each
A ∈ Sp (2k) denote by GrA the graph of A which is a Lagrangian subspace
of the symplectic vector space R4k = R2k×R2k equipped with the symplectic
structure ω4k = −ω2k ⊕ ω2k. Denote by ∆ the diagonal in R
4k = R2k × R2k
– it is a Lagrangian subspace with respect to ω4k. Now for any smooth path
{At}, 0 ≤ t ≤ 1, in Sp (2k) define CZmatr as
CZmatr({At}) := RS({GrAt},∆).
Equivalently, one can define CZmatr({At}) similarly to the index RS by look-
ing at the intersections of {A(t)} with the hypersurface Σ ⊂ Sp (2k) formed
by all the symplectic 2k× 2k-matrices with eigenvalue 1 and translating the
notions of a regular crossing and the corresponding quadratic form to this
setup.
Both indices Ind2k ({At}, V ) and CZmatr({At}) depend only on the fixed
endpoints homotopy class of the path {At} and are additive with respect to
the concatenation of paths in Sp (2k). The relation between the two indices
is as follows. Denote by I2k the 2k × 2k identity matrix. Given a smooth
path {At}, 0 ≤ t ≤ 1, in Sp (2k), set Ât := I2k ⊕ At ∈ Sp (4k). Then
CZmatr({At}) = Ind4k({Ât},∆). (12)
Remark 3.2. Note that near each W ∈ ΣV there exists a local coordinate
chart (on Lagr (k)) in which ΣV can be defined by an algebraic equation of
degree bounded from above by a constant C depending only on k and W .
Moreover, since for any two V, V ′ ∈ Lagr (k) there exists a diffeomorphism of
Lagr (k) mapping ΣV into ΣV ′ we can assume that C = C(k) is independent
of W and depends only on k. Therefore for any V , for any point W ∈ ΣV
and for any sufficiently small open neighborhood UW of W in Lagr (k) the
number of connected components of UW \(UW ∩ΣV ) is bounded by a constant
depending only on k.
Using these observations and the fact that regular crossings are isolated it
is easy to show that there exists a constant C(k), depending only on k, such
that for any Lagrangian subspace V ⊂ R2k and any path {At} ⊂ Sp (2k),
0 ≤ t ≤ 1, there exists a δ > 0 such that for any smooth path {A′t} ⊂ Sp (2k),
0 ≤ t ≤ 1, which is δ-close to {At} in the C
0-metric, one has
|Ind2k({At}, V )− Ind2k({A
t}, V | < C(k),
|CZmatr({At})− CZmatr({A
t}| < C(k).
Leray theorem on the index Ind2k: The following result follows from
Theorem 5.1 in [54] which Robbin and Salamon credit to Leray [35], p.52.
Denote by L the Lagrangian (q1, . . . , qk)-coordinate plane in R
2k. Any sym-
plectic matrix S ∈ Sp (2k) can be decomposed into k × k blocks as
where the blocks satisfy, in particular, the condition that
EF T − FET = 0. (13)
If SL ∩ L = 0 then the k × k-matrix F is invertible and multiplying (13)
by F−1 on the left and (F T )−1 = (F−1)T on the right, we get that F−1E −
ET (F−1)T = 0. Therefore the matrix QS := F
−1E is symmetric.
Theorem 3.3 ([54], Theorem 5.1; [35], p.52). Assume {At}, {Bt}, 0 ≤ t ≤ 1,
are two smooth paths in Sp (2k), such that A0 = B0 = I2k and A1L ∩ L = 0,
B1L ∩ L = 0, A1B1L ∩ L = 0. Then
Ind2k({AtBt}, L) = Ind2k({At}, L) + Ind2k({Bt}, L) +
sign (QA1 +QB1),
where sign (QA1 +QB1) is the signature of the quadratic form defined by the
symmetric k × k-matrix QA1 +QB1.
Corollary 3.4. Let V be any Lagrangian subspace of R2k. Then there exists
a positive constant C, depending only on k, such that for any smooth paths
{Xt}, {Yt}, 0 ≤ t ≤ 1, in Sp (2k), such that X0 = Y0 = I2k (there are no
assumptions on X1, Y1!),
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| < C.
Proof. We will write C1, C2, . . . for (possibly different) positive constants de-
pending only on k.
Pick a map Ψ ∈ Sp (2k) such that ΨV = L. Denote At = ΨXtΨ
Bt = ΨYtΨ
−1. Note that the paths {At}, {Bt} are based at the identity.
Using the naturality property (11) of Ind2k we get
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| =
= |Ind2k({ΨXtYtΨ
−1},ΨV )− Ind2k({ΨXtΨ
−1},ΨV )−
−Ind2k({ΨYtΨ
−1},ΨV )| =
= |Ind2k({(ΨXtΨ
−1)(ΨYtΨ
−1)}, L)− Ind2k({ΨXtΨ
−1}, L)−
−Ind2k({ΨYtΨ
−1}, L)| =
= |Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|.
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| =
= |Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|. (14)
Further on, Remark 3.2 implies that we can find sufficiently C0-close identity-
based perturbations {A′t}, {B
t} of {At}, {Bt} such that
A′1L ∩ L = 0, B
1L ∩ L = 0, A
1L ∩ L = 0. (15)
|Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|−
−|Ind2k({A
t}, L)− Ind2k({A
t}, L)− Ind2k({B
t}, L)| < C1, (16)
for some C1. On the other hand, since the three identity-based paths {A
{B′t}, {A
t}, satisfy the conditions (15), we can apply to them Theorem 3.3.
Hence there exists C2 such that
|Ind2k({A
t}, L)− Ind2k({A
t}, L)− Ind2k({B
t}, L)| < C2.
Combining it with (14) and (16) we get that there exists C3 such that
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| < C3,
which finishes the proof.
Conley-Zehnder index as a quasi-morphism: Recall that 2n = dimM .
Restricting CZmatr to the identity-based paths in Sp (2n) one gets a function
on S̃p (2n) that will be still denoted by CZmatr.
Proposition 3.5 (cf. [19]). The function CZmatr : S̃p (2n) → R is a quasi-
morphism. It means that there exists a constant C > 0 such that
|CZmatr(ab)− CZmatr(a)− CZmatr(b)| ≤ C ∀a, b ∈ S̃p (2n).
Proof. Represent a and b by identity-based paths {At}, {Bt}, 0 ≤ t ≤ 1, in
Sp (2n). Then use (12) and apply Corollary 3.4 for k = 2n, V = ∆ to {Ât},
{B̂t} in Sp (4n).
Maslov index of symplectic loops: The Conley-Zehnder index for
identity-based loops in Sp (2n) is called the Maslov index of a loop. Its
original definition, going back to [4], is the following: it is the intersection
number of an identity-based loop with the stratified hypersurface Σ whose
principal stratum is equipped with a certain co-orientation. Note that we do
not divide the intersection number by 2 and thus in our case the Maslov index
takes only even values; for instance, the Maslov index of a counterclockwise
2π-twist of the standard symplectic R2 is 2. We denote the Maslov index of
a loop {B(t)} by Maslov ({B(t)}).
Conley-Zehnder and Maslov indices of periodic orbits: The Con-
ley-Zehnder index for periodic orbits is defined by means of the Conley-
Zehnder index for matrix paths as follows. Given [γ, u] ∈ P̃F , build an
identity-based path {A(t)} in Sp (2n) as follows: take a symplectic trivial-
ization of the bundle u∗(TM) over D2 and use the trivialization to identify
the linearized flow dγ(0)ft, 0 ≤ t ≤ 1, along γ with a symplectic matrix
{A(t)}. Then the Conley-Zehnder index CZF ([γ, u]) is defined as
CZF ([γ, u]) := n− CZmatr ({A(t)}). (17)
With such a normalization of CZF for any sufficiently C
2-small autonomous
Morse Hamiltonian F , the Conley-Zehnder index of an element of P̃F , rep-
resented by a pair [x, u] consisting of a critical point x of F (viewed as a
constant path in M) and the trivial disk u, is equal to the Morse index of
x. Note that with such a normalization CZF (Sy) = CZF (y)+2
c1(M) for
every y ∈ P̃F and S ∈ H
2 (M).
Similarly, if the time-1 flow generated by F defines a loop in Ham(M) then
to each [γ, u] ∈ P̃F one can associate its Maslov index. Namely, trivialize the
bundle u∗(TM) over D2 and identify the linearized flow {dxft} along γ with
an identity-based loop of symplectic 2n × 2n-matrices. Define the Maslov
index mF ([γ, u]) as the Maslov index for the loop of symplectic matrices.
Under the action of HS2 (M) on P̃F the Maslov index changes as follows:
mF (S · [γ, u]) = mF ([γ, u])− 2
c1(M), S ∈ H
2 (M).
Let us make the following remark. Assume γ ∈ PF and assume that a
symplectic trivialization of the bundle γ∗(TM) over S1 identifies {dγ(0)ft}
with an identity-based path {A(t)} of symplectic matrices. Assume there
is another symplectic trivialization of the same bundle, coinciding with the
first one at γ(0), and denote by {B(t)} the identity-based loop of transition
matrices from the first symplectic trivialization to the second one. Use the
second trivialization to identify {dγ(0)ft} with an identity-based path {A
′(t)}.
CZmatr ({A
′(t)}) = CZmatr ({A(t)}) +Maslov ({B(t)}), (18)
and if {A(t)} is a loop then so is {A′(t)} and
Maslov ({A′(t)}) = Maslov ({A(t)}) +Maslov ({B(t)}). (19)
3.4 Spectral numbers
Given the algebraic setup as above, the construction of the Piunikhin-Sala-
mon-Schwarz (PSS) isomorphism [47] yields a Λ-linear isomorphism (PSS-
isomorphism) φM : QH∗(M) → HF∗(F, J) which preserves the grading and
which is actually a ring isomorphism (the pair-of-pants product defines a ring
structure on HF∗(F, J)).
Using the PSS-isomorphism one defines the spectral numbers c(a, F ),
where 0 6= a ∈ QH∗(M), in the usual way [45]. Namely, the action functional
AF defines a filtration on C(F ) which induces a filtration HF
∗ (F, J), α ∈ R,
on HF∗(F, J), with HF
∗ (F, J) ⊂ HF
∗ (F, J) as long as α < β. Then
c(a, F ) := inf {α | φM(a) ∈ HF
∗ (F, J)}.
Such spectral number is finite and well-defined (does not depend on J). Here
is a brief account of the relevant properties of spectral numbers – for details
see [45] (see also [65, 42, 59, 43] for earlier versions of this theory).
(Spectrality) c(a,H) ∈ spec (H), where the spectrum spec (H) of H is
defined as the set of critical values of the action functional AH , i.e.
spec (H) := AH(P̃H) ⊂ R.
(Quantum homology shift property) c(λa,H) = c(a,H) + ν(λ) for all
λ ∈ Λ, where ν is the valuation defined in Section 3.1.
(Hamiltonian shift property) c(a,H + λ(t)) = c(a,H) +
λ(t) dt for
any Hamiltonian H and function λ : S1 → R.
(Monotonicity) If H1 ≤ H2, then c(a,H1) ≤ c(a,H2).
(Lipschitz property) The map H 7→ c(a,H) is Lipschitz on the space of
(time-dependent) Hamiltonians H : M × S1 → R with respect to the
C0-norm.
(Symplectic invariance) c(a, φ∗H) = c(a,H) for every φ ∈ Symp0(M),
H ∈ C∞(M); more generally, Symp (M) acts on H∗(M ;F), and hence
on QH∗(M), and c(a, φ
∗H) = c(φ∗a,H) for any φ ∈ Symp (M).
(Normalization) c(a, 0) = ν(a) for every a ∈ QH∗(M).
(Homotopy invariance) c(a,H1) = c(a,H2) for any normalized H1, H2
generating the same φ ∈ H̃am (M). Thus one can define c(a, φ) for any
φ ∈ H̃am (M) as c(a,H) for any normalized H generating φ.
(Triangle inequality) c(a ∗ b, φψ) ≤ c(a, φ) + c(b, ψ).
The commutative ring QH•(M) admits a K-bilinear and K-valued form Ω
on QH•(M) which associates to a pair of quantum homology classes a, b ∈
QH•(M) the coefficient (belonging to K) at the class [point] = [point] · q
a point in their quantum product a ∗ b ∈ QH•(M) (the Frobenius structure).
Let τ : K → F be the map sending each series
θ∈Γ zθs
θ, zθ ∈ F , to its free
term z0. Define a non-degenerate F -valued F -linear pairing on QH•(M) by
Π(a, b) := τΩ(a, b) = τΩ(a ∗ b, [M ]) . (20)
Note that Π is symmetric and
Π(a ∗ b, c) = Π(a, b ∗ c) ∀a, b, c ∈ QH•(M). (21)
With this notion at hand, we can present another important property of
spectral numbers:
(Poincaré duality) c(b, φ) = − infa∈Υ(b) c(a, φ
−1) for all b ∈ QH•(M)\{0}
and φ. Here Υ(b) denotes the set of all a ∈ QH•(M) with Π(a, b) 6= 0.
The Poincaré duality can be extracted from [47] (cf. [22]) – for a proof see
[46].
The next property is an immediate consequence of the definitions (see [22]
for a discussion in the monotone case):
(Characteristic exponent property) Given 0 6= λ ∈ F , a, b ∈ QH∗(M),
a, b, a + b 6= 0, and a (time-dependent) Hamiltonian H , one has
c(λ · a,H) = c(a,H) and c(a+ b,H) ≤ max(c(a,H), c(b,H)).
3.5 Partial symplectic quasi-states
Given a non-zero idempotent a ∈ QH2n(M) and a time-independent Hamil-
tonian H :M → R, define
ζ(a,H) := lim
c(a, lH)
. (22)
When a is fixed, we shall often abbreviate ζ(H) instead of ζ(a,H). The limit
in the formula (22) always exists and thus the functional ζ : C∞(M) → R is
well-defined. The functional ζ on C∞(M) is Lipschitz with respect to the C0-
norm ‖H‖ = maxM |H| and therefore extends to a functional ζ : C(M) → R,
where C(M) is the space of all continuous functions on M . These facts were
proved in [23] in the case a = [M ] but the proofs actually go through for any
non-zero idempotent a ∈ QH2n(M).
Here we will list the properties of ζ for such an M . Again, these proper-
ties were proved in [23] in the case a = [M ] but the proof goes through for
any non-zero idempotent a ∈ QH2n(M). The additivity with respect to con-
stants property was not explicitly listed in [23] but follows immediately from
the definition of ζ and the Hamiltonian shift property of spectral numbers.
The triangle inequality follows readily from the definition of ζ and from the
triangle inequality for the spectral numbers.
Theorem 3.6. The functional ζ : C(M) → R satisfies the following prop-
erties:
Semi-homogeneity: ζ(αF ) = αζ(F ) for any F and any α ∈ R≥0.
Triangle inequality: If F1, F2 ∈ C
∞(M), {F1, F2} = 0 then ζ(F1 + F2) ≤
ζ(F1) + ζ(F2).
Partial additivity and vanishing: If F1, F2 ∈ C
∞(M), {F1, F2} = 0 and the
support of F2 is displaceable, then ζ(F1 + F2) = ζ(F1); in particular, if the
support of F ∈ C(M) is displaceable, ζ(F ) = 0.
Additivity with respect to constants and normalization: ζ(F +α) = ζ(F )+α
for any F and any α ∈ R. In particular, ζ(1) = 1.
Monotonicity: ζ(F ) ≤ ζ(G) for F ≤ G.
Symplectic invariance: ζ(F ) = ζ(F ◦ f) for every symplectic diffeomorphism
f ∈ Symp0 (M).
Characteristic exponent property: ζ(a1+a2, F ) ≤ max(ζ(a1, F ), ζ(a2, F )) for
each pair of non-zero idempotents a1, a2 with a1 ∗ a2 = 0, a1+ a2 6= 0 (in this
case a1 + a2 is also a non-zero idempotent), and for all F ∈ C(M) .
We will call the functional ζ : C(M) → R satisfying all the properties
listed in Theorem 3.6 a partial symplectic quasi-state.
4 Basic properties of (super)heavy sets
In this section we prove parts (i) and (iii) of Theorem 1.2, as well as The-
orem 1.3. We shall use that a partial symplectic quasi-state ζ extends by
continuity in the uniform norm to a monotone functional on the space of
continuous functions C(M), see Section 3.5 above. In particular, one can
use continuous functions instead of the smooth ones in the definition of (su-
per)heaviness in formulae (3) and (4).
Assume a partial quasi-state ζ defined by a non-zero idempotent is fixed
and we consider heaviness and superheaviness with respect to ζ . We start
with the following elementary
Proposition 4.1. A closed subset X ⊂ M is heavy if and only if for every
H ∈ C∞(M) with H|X = 0, H ≤ 0 one has ζ(H) = 0. A closed subset
X ⊂ M is superheavy if and only if for every H ∈ C∞(M) with H|X = 0,
H ≥ 0 one has ζ(H) = 0.
Proof. The “only if” parts follow readily from the monotonicity property of
ζ . Let us prove the “if” part in the “heavy case” – the “superheavy” case is
similar. Take a function H on M and put
F = min(H − inf
H, 0) .
Note that F |X = 0 and F ≤ 0. Thus ζ(F ) = 0 by the assumption of the
proposition. Thus
0 = ζ(F ) ≤ ζ(H − inf
H) = ζ(H)− inf
which yields heaviness of X .
The following proposition proves part (i) of Theorem 1.2.
Proposition 4.2. Every superheavy set is heavy.
Proof. Let X ⊂ M be a superheavy subset. Assume that H|X = 0, H ≤ 0.
By the triangle inequality for ζ we have ζ(H) + ζ(−H) ≥ 0. Note that
−H|X = 0, −H ≥ 0. Superheaviness yields ζ(−H) = 0, so ζ(H) ≥ 0. But
by monotonicity ζ(H) ≤ 0. Thus ζ(H) = 0 and the claim follows from
Proposition 4.1.
Superheavy sets have the following user-friendly property.
Proposition 4.3. Let X ⊂ M be a superheavy set. Then for every α ∈ R
and H ∈ C∞(M) with H|X ≡ α one has ζ(H) = α.
Proof. Since ζ(H + α) = ζ(H) + α it suffices to prove the proposition for
α = 0. Take any function H with H|X = 0. Since X is superheavy and, by
Proposition 4.2, also heavy, we have
0 = ζ(−|H|) ≤ ζ(H) ≤ ζ(|H|) = 0 ,
which yields ζ(H) = 0.
As an immediate consequence we get part (iii) of Theorem 1.2.
Proposition 4.4. Every superheavy set intersects with every heavy set.
Proof. Let X be a superheavy set and Y be a heavy set. Assume on the
contrary that X ∩ Y = ∅. Take a function H ≤ 0 with H|Y ≡ 0 and
H|X ≡ −1. Then ζ(H) = −1 by Proposition 4.3. On the other hand,
ζ(H) = 0 since Y is heavy, and we get a contradiction.
Note that two heavy sets do not necessarily intersect each other: a meridian
of T2 is heavy (see Corollary 6.4 below), while two meridians can be disjoint.
Proof of Theorem 1.3 (i) and (ii): The triangle inequality yields
c(a,H) = c(a ∗ [M ], 0 +H) ≤ c(a, 0) + c([M ], H) = ν(a) + c([M ], H).
Passing to the partial quasi-states ζ(a,H) and ζ([M ], H) we get:
ζ(a,H) = lim
c(a, kH)/k ≤
≤ lim
(ν(a) + c([M ], kH))/k = lim
c([M ], kH)/k = ζ([M ], H).
The result now follows from the definition of heavy and superheavy sets (see
Definition 1.1).
Proof of Theorem 1.3 (iii): By the characteristic exponent property of
spectral invariants,
ζ(a, F ) ≤ max
i=1,...,l
ζ(ei, F ) ∀F ∈ C
∞(M) . (23)
Choose a sequence of functions Gj ∈ C
∞(M), j → +∞, with the fol-
lowing properties: Gk ≤ Gj for k > j, Gj = 0 on X , Gj ≤ 0 and for
every function F ≤ 0 which vanishes on an open neighborhood of X there
exists j so that Gj ≤ F (existence of such a sequence can be checked easily).
In view of inequality (23), we have that for every j there exists i so that
ζ(a,Gj) ≤ ζ(ei, Gj). Passing, if necessary, to a subsequence Gjk , jk → +∞,
we can assume without loss of generality that i is the same for all j. In view
of heaviness of X with respect to a, we have that ζ(a,Gj) = 0. Therefore
ζ(ei, Gj) ≥ 0.
Choose any function F ≤ 0 onM which vanishes on an open neighborhood
of X . Then there exists j large enough so that F ≥ Gj. By monotonicity
combined with the previous estimate we have
0 ≥ ζ(ei, F ) ≥ ζ(ei, Gj) ≥ 0 ,
which yields ζ(ei, F ) = 0.
Now let F be any continuous function on M that vanishes on X . Take
a sequence of continuous functions Fj , converging to F in the C
0-norm, so
that each Fj vanishes on an open neighborhood of X . Then ζ(ei, Fj) =
limj→+∞ ζ(ei, Fj) = 0, because ζ(ei, ·) is Lipschitz with respect to the C
norm. The heaviness ofX with respect to ei now follows from Proposition 4.1.
This finishes the proof of the theorem.
5 Products of (super)heavy sets
In this section we prove Theorem 1.5 on products of (super)heavy subsets.
5.1 Product formula for spectral invariants
The proof of Theorem 1.5 is based on the following general result.
Theorem 5.1. For every pair of time-dependent Hamiltonians G1, G2 onM1
and M2, and all non-zero a1 ∈ QHi1(M1), a2 ∈ QHi2(M2) we have
c(a1 ⊗ a2, G1(z1, t) + G2(z2, t)) = c(a1, G1) + c(a1, G2) .
Here G1(z1, t) +G2(z2, t) is a time-dependent Hamiltonian on M1 ×M2.
Let us deduce Theorem 1.5 from Theorem 5.1.
Proof of Theorem 1.5: We show that the product of superheavy sets is
superheavy (the proof for heavy sets goes without any changes). We denote
by ζ1, ζ2 and ζ the partial quasi-states on M1,M2 and M := M1 ×M2 as-
sociated to the idempotents a1, a2 and a1 ⊗ a2 respectively. Let Xi ⊂ Mi,
i = 1, 2, be a superheavy set. By Proposition 4.1 it suffices to show that if a
non-negative function G ∈ C∞(M) vanishes on some neighborhood, say U ,
of X := X1 × X2 then ζ(G) = 0. (Since ζ is Lipschitz with respect to the
C0-norm this would imply that ζ(G) = 0 for any non-negative G ∈ C(M)
that vanishes on X). Put K := maxM G. Choose neighborhoods Ui of Xi so
that U1 ×U2 ⊂ U . Choose non-negative functions Gi on Mi which vanish on
Xi and such that Gi(z) > K for all z ∈Mi \Ui. Observe that G ≤ G1 +G2.
But, in view of Theorem 5.1 and superheaviness of Xi, we have
ζ(G1 +G2) = ζ1(G1) + ζ2(G2) = 0 .
By monotonicity
0 ≤ ζ(G) ≤ ζ(G1 +G2) = 0 ,
and thus ζ(G) = 0.
It remains to prove Theorem 5.1. Note that the left-hand side of the equality
stated in the theorem does not exceed the right-hand side: this is an imme-
diate consequence of the triangle inequality for spectral invariants. However,
we were unable to use this observation for proving the theorem. Our ap-
proach is based on a rather lengthy algebraic analysis which enables us to
calculate separately the left and the right-hand sides “on the chain level”. A
simple inspection of the results of this calculation yields the desired equality.
5.2 Decorated Z2-graded complexes
A Z2-complex is a Z2-graded finite-dimensional vector space V over a field
K equipped with a K-linear differential ∂ : V → V satisfying ∂2 = 0 and
shifting the grading. A decorated complex over K = KΓ includes the following
data:
• a countable subgroup Γ ⊂ R;
• a Z2-graded complex (V, d) over KΓ;
• a preferred basis x1, . . . , xn of V ;
• a function F : {x1, . . . , xn} → R (called the filter) which extends to V
λjxj) = max{ν(λj) + F (xj)
∣∣∣ λj 6= 0},
and satisfies F (dv) < F (v) for all v ∈ V \ {0}. The convention is that
F (0) = −∞. Here ν is the valuation defined in Section 3.1 above.
We shall use the notation
V := (V, {xi}i=1,...,n, F, d,Γ)
for a decorated complex.
The ⊗̂K-tensor product V = V1⊗̂KV2 of decorated complexes
Vi = (Vi, {x
j }j=1,...,ni, Fi, di,Γi) , i = 1, 2
is defined as follows. Consider the space V = V1⊗̂KV2 (see formula (5) above)
with the natural Z2-grading. Define the differential d on V by
d(x⊗ y) = d1x⊗ y + (−1)
deg xx⊗ d2y .
The preferred basis in V is given by {xpq := x
p ⊗ x
q } and the filter F is
defined by
F (xpq) = F1(x
p ) + F2(x
Finally, we put V := (V, {xpq}, F, d,Γ1 + Γ2) .
The (Z2-graded) homology of decorated complexes are denoted by H∗(V)
– they are K-vector spaces. By the Künneth formula, H(V1⊗̂KV2) =
H(V1)⊗̂KH(V2).
Next we define spectral invariants associated to a decorated complex V :=
(V, {xpq}, F, d) . Namely, for a ∈ H(V) put
c(a) := inf{F (v) | a = [v], v ∈ Ker d} .
We shall see below that c(a) > −∞ for each a 6= 0.
The purpose of this algebraic digression is to state the following result:
Theorem 5.2. For any two decorated complexes V1,V2
c(a1 ⊗ a2) = c(a1) + c(a2) ∀a1 ∈ H(V1), a2 ∈ H(V2)
5.3 Reduced Floer and Quantum homology
The 2-periodicity of the Floer complex and Floer homology defined by the
multiplication by q (see Proposition 3.1 above) allows to encode their al-
gebraic structure in a decorated Z2-complex. Consider a regular pair (G, J)
consisting of a Hamiltonian function and a compatible almost-complex struc-
ture on M (both, in general, are time-dependent). Let (C∗(G), dG,J) be the
corresponding Floer complex. Let us associate to it a Z2-complex: a Z2-
graded vector space VG over KΓ, defined as
VG := C0(G)⊕ C1(G),
with the obvious Z2-grading, and a differential ∂G,J : VG → VG, defined as
the direct sum of dG,J : C1(G) → C0(G) and qdG,J : C0(G) → C1(G). One
readily checks that this is indeed a Z2-complex because dG,J : C(G) → C(G)
is ΛΓ-linear. We will call (VG, ∂G,J) the Z2-complex associated to (G, J).
Note that the cycles and the boundaries of (VG, ∂G) having Z2-degree
i ∈ {0, 1} in VG coincide, respectively, with the cycles and the boundaries
having Z-degree i of (C(G), dG,J). Therefore the Floer homology HFi(G, J)
is isomorphic, as a vector space over KΓ, to the i-th degree component of the
homology of the complex (VG, ∂G,J).
The Z2-complex (VG, ∂G,J) carries a structure of the decorated complex
VG,J as follows. Let γi(t), i = 1, . . . , m, be the collection of all contractible
1-periodic orbits of the Hamiltonian flow generated by G. Choose disc ui
in M spanning γi. For each i there exists unique integer, say ri, so that
the Conley-Zehnder index of the element xi := q
ri · [γi, ui] lies in the set
{0, 1}. Clearly, the collection {xi} forms a basis of VG over KΓ. We shall
consider it as a preferred basis. Note that the preferred basis is unique up to
multiplication of xi’s by elements of the form s
αi , αi ∈ Γ. Finally, the action
functional associated to G defines a filtration on VG.
The homology of (VG, ∂G,J) can be canonically identified via the PSS-
isomorphism with the object which we call reduced quantum homology:
QHred(M) := QH0(M)⊕QH1(M) .
We call this isomorphism the reduced PSS-isomorphism and denote it by ψG,J .
Note that we have a natural projection p : QH∗(M) → QHred(M) which
sends any degree homogeneous element a to aqr with deg a + 2r ∈ {0, 1}.
With this notation, the usual Floer-homological spectral invariant c(a,G)
coincides with the spectral invariant c(p(a)) of the decorated complex VG,J .
5.4 Proof of Theorem 5.1
By the Lipschitz property of spectral numbers it is enough to consider the
case when G1 and G2 belong to regular pairs (Gi, Ji), i = 1, 2. Set
G(z1, z2, t) := G1(z1, t) +G(z2, t)
and J := J1 × J2. Then (G, J) is also a regular pair. Put Γi = Γ(Mi, ωi). It
is straightforward to see that the decorated complex VG,J is the ⊗̂K-tensor
product of the decorated complexes VGi,Ji for i = 1, 2.
Put (M,ω) = (M1×M2, ω1⊕ω2). An obvious modification of the Künneth
formula for quantum homology (see e.g. [41, Exercise 11.1.15] for the state-
ment in the monotone case) yields a natural monomorphism
ı : QHi1(M1, ω1)⊗̂KQHi2(M1, ω1) → QHi1+i2(M,ω) .
Since in our setting quantum homologies are 2-periodic, the collection of these
isomorphisms for all pairs (i1, i2) from the set {0, 1} induces an isomorphism
j : QHred(M1)⊗̂KQHred(M2) → QHred(M) .
It has the following properties: First, given two elements a1 ∈ QHi1(M1, ω1)
and a2 ∈ QHi2(M2, ω2) we have that
p(a1)⊗ p(a2) = p(a1 ⊗ a2) .
Second, the following diagram commutes:
H(VG1, ∂G1,J1)⊗̂KH(VG2 , ∂G2,J2)
ψG1,J1⊗ψG2,J2
H(VG, ∂G,J)
QHred(M1)⊗̂KQHred(M2)
// QHred(M)
Here k is the isomorphism coming from the Künneth formula for Z2-comple-
xes, and ψGi,Ji, ψG,J stand for the reduced PSS-isomorphisms. It follows that
the definition of c(ai, Gi), c(a1⊗a2, G) matches the definition of c(p(ai)) and
c(p(a1)⊗ p(a2)). By Theorem 5.2 we get that
c(a1⊗a2, G) = c(p(a1)⊗p(a2)) = c(p(a1))+ c(p(a2)) = c(a1, G1)+ c(a2, G2) .
This proves Theorem 5.1 modulo Theorem 5.2.
5.5 Proof of algebraic Theorem 5.2
A decorated complex is called generic if F (xi) − F (xj) /∈ Γ for all i 6= j
(recall that under our assumptions Γ, the group of periods of the symplectic
form ω over π2(M), is a countable subgroup of R). We start from some
auxiliary facts from linear algebra. Let V := (V, {xi}i=1,...,n, F, d,Γ) be a
generic decorated complex. We recall once again that for brevity we write K
instead of KΓ wherever it is clear what Γ is taken.
An element x ∈ V is called normalized if
x = xp +
i 6=p
λixi , λi ∈ K, F (xp) > max
i 6=p
F (λixi) .
We shall use the notation x = xp+o(xp). In generic complexes, every element
x 6= 0 can be uniquely written as x = λ(xp+o(xp)) for some p = 1, . . . , n and
λ ∈ K. A system of vectors e1, . . . , em in V is called normal if every ei has
the form ei = xji +o(xji) for ji ∈ {1, . . . , n} and the numbers ji are pair-wise
distinct.
Lemma 5.3. Let e1, . . . , em be a normal system. Then
λiei) = max
F (λiei) .
Proof. We prove the result using induction in m. For m = 1 the statement is
obvious. Let’s check the induction step m− 1 → m. Observe that it suffices
to check that
F (e1 +
λiei) ≥ F (e1) . (24)
Then obviously
λiei) ≥ max
F (λiei) ,
while the reversed inequality is an immediate consequence of the definitions.
By the induction step,
λiei) = max
i=2,...,n
F (λiei) .
In view of the genericity, the maximum at the right hand side can be uniquely
written as F (λi0xi0). Without loss of generality we shall assume that ei =
xi + o(xi) and i0 = 2.
λ−12 λiei = x2 + o(x2) .
Write
e1 = x1 + αx2 +X, v = x2 + βx1 + Y,
where α, β ∈ K and X, Y ∈ SpanK(x3, . . . , xn). Note that F (x1) > F (αx2),
F (x2) > F (βx1), which yields
ν(α) < F (x1)− F (x2) < −ν(β) = ν(β
−1) . (25)
In particular, ν(α) < ν(β−1). Note that
e1 + λ2v = (1 + λ2β)x1 + (α + λ2)x2 + Z, Z ∈ SpanK(x3, . . . , xn) .
F (e1 + λ2v) ≥ max(ν(1 + λ2β) + F (x1), ν(α + λ2) + F (x2)) .
If ν(1 + λ2β) ≥ 0 we have F (e1 + λ2v) ≥ F (x1) = F (e1) and inequality (24)
follows. Assume that ν(1+λ2β) < 0 = ν(1). Then ν(λ2β) = 0 = ν(λ2)+ν(β),
and hence ν(λ2) = ν(β
−1) 6= ν(α). Thus
ν(α + λ2) ≥ ν(λ2) = −ν(β) .
Combining this inequality with (25) we get that
F (e1 + λ2v) ≥ ν(α + λ2) + F (x1) + (F (x2)− F (x1))
≥ F (x1) + (ν(α + λ2) + ν(β)) ≥ F (x1) = F (e1) .
This completes the proof of inequality (24), and hence of the lemma.
It readily follows from the lemma that every normal system is linearly inde-
pendent.
Lemma 5.4. Every subspace L ⊂ V has a normal basis.
Proof. We use induction over m = dimK L. The case m = 1 is obvious,
so let us handle the induction step m − 1 → m. It suffices to show the
following: Let e1, . . . , em−1 be a normal basis in a subspace L
′, and let v /∈ L′
be any vector. Put L = SpanK(L
′ ∪ {v}). Then there exists em ∈ L so that
e1, . . . , em is a normal basis. Indeed, assume without loss of generality that
for all i = 1, . . . , m−1 one has ei = xi+ o(xi). Put W = SpanK(xm, . . . , xn).
We claim that L′ ∩W = {0}. Indeed, otherwise
λ1e1 + . . .+ λm−1em−1 = λmxm + . . .+ λnxn
where the linear combinations in the right and the left-hand sides are non-
trivial. Apply F to both sides of this equality. By Lemma 5.3
F (λ1e1 + . . .+ λm−1em−1) = F (xp) mod Γ, where 1 ≤ p ≤ m− 1 ,
while
F (λmxm + . . .+ λnxn) = F (xq) mod Γ, where q ≥ m .
This contradicts the genericity of our decorated complex, and the claim fol-
lows. Since dimL′+dimW = dimV , we have that V = L′⊕W . Decompose v
as u+w with u ∈ L′, w ∈ W , and note that w ∈ L. Note that e1, . . . , em−1, w
are linearly independent. Furthermore, w = λ(xp + o(xp)) for some p ≥ m.
Put em = λ
−1w. The vectors e1, . . . , em form a normal basis in L.
The same proof shows that if L1 ⊂ L2 are subspaces of V , every normal basis
in L1 extends to a normal basis in L2.
Now we turn to the analysis of the differential d. Choose a normal basis
g1, . . . , gq in Im d, and extend it to a normal basis g1, . . . , gq, h1, . . . , hp in
Ker d. Note that each of these p + q vectors has the form xj + o(xj) with
distinct j. Let us assume without loss of generality that the remaining n−p−q
elements of the preferred basis in V are x1, . . . , xq, and
gi = xi+q + o(xi+q), hj = xj+2q + o(xj+2q) .
Here we use that, by the dimension theorem, n = p+ 2q. Note that
x1, . . . , xq, g1, . . . , gq, h1, . . . , hp
is a normal system, and hence a basis in V . We call such a basis a spectral
basis of the decorated complex V.
Note that [h1], . . . , [hp] is a basis in the homology H(V). Consider any
homology class a =
λi[hi]. Every element v ∈ V with a = [v] can be
written as v =
λihi +
αjgj. Thus, by Lemma 5.3, F (v) ≥ maxi F (λihi)
and hence
c(a) = max
F (λihi) . (26)
This proves in particular that the spectral invariants are finite provided a 6= 0.
For finite sets A = {v1, . . . , vs} and B = {w1, . . . , ws} we write A⊗B for the
finite set {vi ⊗ wj}.
Assume now that V1,V2 are generic decorated complexes. We say that they
are in general position if their tensor product V = V1⊗̂KV2 is generic. Let
Bi = {x
1 , . . . , x
1 , . . . , g
1 , . . . , h
}, i = 1, 2
be a spectral basis in Vi. Obviously, B1 ⊗ B2 is a normal basis in V1⊗̂KV2.
We shall denote by d1, d2, d the differentials and by F1, F2, F the filters in
V1,V2 and V respectively. Put Gi = {g
1 , . . . , g
qi }, Hi = {h
1 , . . . , h
and K = G1 ⊗ B2 ∪B1 ⊗G2. Observe that
Im d ⊂W := Span(K) .
Take any two classes
j ] ∈ H(Vi) , i = 1, 2.
Suppose that a1 ⊗ a2 = [v]. Then v is of the form
λ(1)m λ
m ⊗ h
l + w
where w must lie in W . Observe that (H1 ⊗H2) ∩K = ∅. By Lemma 5.3,
F (v) ≥ max
F (λ(1)m λ
m ⊗ h
l ) ,
and hence
c(a1 ⊗ a2) = max
F (λ(1)m λ
m ⊗ h
= max
m ) + F2(λ
= max
m ) + max
l ) = c(a1) + c(a2) .
In the last equality we used (26). This completes the proof of Theorem 5.2
for decorated complexes in general position.
It remains to remove the general position assumption. This will be done
with the help of the following lemma. We shall work with a family of deco-
rated complexes
V := (V, {xi}i=1,...,n, F, d,Γ)
which have exactly the same data (preferred basis, grading, differential and
Γ) with the exception of the filter F which will be allowed to vary in the class
of filters. The corresponding spectral invariants will be denoted by c(a, F ).
Lemma 5.5.
(i) If filters F, F ′ satisfy F (xi) ≤ F
′(xi) for all i = 1, . . . , n, then c(a, F ) ≤
c(a, F ′) for all non-zero classes a ∈ H(V).
(ii) If F is a filter and θ ∈ R, then F + θ is again a filter and c(a, F + θ) =
c(a, F ) + θ for all non-zero classes a ∈ H(V).
The proof is obvious and we omit it. It follows that for any two filters F, F ′
|c(a, F )− c(a, F ′)| ≤ ||F − F ′||C0 ∀a ∈ H(V) \ {0} .
Assume now that V1,V2 are decorated complexes. Denote by F1, F2 their
filters. Fix ǫ > 0. By a small perturbation of the filters we get new filters,
F ′1 and F
2, on our complexes so that the complexes become generic and in
general position, and furthermore
||F1 − F
1||C0 ≤ ǫ , ||F2 − F
2||C0 ≤ ǫ .
Given homology classes ai ∈ H(Vi) we have
|c(a1, F1) + c(a2, F2)− c(a1 ⊗ a2, F1 + F2)| ≤
|c(a1, F
1) + c(a2, F
2)− c(a1 ⊗ a2, F
1 + F
2)|+ 4ǫ = 4ǫ .
Here we used that Theorem 5.2 is already proved for generic complexes in
general position. Since ǫ > 0 is arbitrary, we get that
c(a1, F1) + c(a2, F2)− c(a1 ⊗ a2, F1 + F2) = 0 ,
which completes the proof of Theorem 5.2 in full generality.
6 Stable non-displaceability of heavy sets
In this section we prove part (ii) of Theorem 1.2.
Proposition 6.1. Every heavy subset is stably non-displaceable.
For the proof we shall need the following auxiliary statement. Given R > 0,
consider the torus T2R obtained as the quotient of the cylinder T
∗S1 = R(r)×
S1 (θ mod 1) by the shift (r, θ) 7→ (r + R, θ). For α > 0 define the function
Fα(r, θ) := αf(r) on T
R, where f(r) is any R-periodic function having only
two non-degenerate critical points on [0, R]: a maximum point at r = 0 with
f(0) = 1, and a minimum point at r = R/2, f(R/2) =: −β < 0. We denote
by [T ] the fundamental class of T2R. We work with the symplectic form dr∧dθ
on T2R.
Lemma 6.2. c([T ], Fα) = α.
Proof. Note that the contractible closed orbits of period 1 of the Hamiltonian
flow generated by Fα are fixed points forming circles S+ = {r = 0} and
S− = {r = R/2}. The actions of the fixed points on S± equal respectively to
α and −αβ, and thus the spectral invariants of Fα lie in the set {α,−αβ}.
Recall from [59] that c([T ], Fα) > c([point], Fα). Thus c([T ], Fα) = α.
Lemma 6.3. Let H ∈ C∞(M) so that H−1(maxH) is displaceable. Then
ζ(H) < maxH.
Proof. Choose ǫ > 0 so that the set
H−1((maxH − ǫ,maxH ])
is displaceable. Choose a real-valued cut-off function ρ : R → [0, 1] which
equals 1 near maxH and which is supported in (maxH−ǫ,maxH+ǫ). Thus
ρ(H) is supported in H−1((maxH − ǫ; maxH ]) and ζ(ρ(H)) = 0. Since H
and ρ(H) Poisson-commute, the vanishing and the monotonicity axioms yield
ζ(H) = ζ(ρ(H)) + ζ(H − ρ(H)) ≤ max(H − ρ(H)) < maxH .
Proof of Proposition 6.1: It suffices to show that for every R > 0 the set
Y := X × {r = 0} ⊂M ′ :=M × T2R
is non-displaceable. Assume on the contrary that Y is displaceable. Choose
a function H on M with H ≤ 0, H−1(0) = X . Put
H ′ = H + F1 = H + f(r) :M
′ → R.
Assume that the partial quasi-state ζ on M is associated to some non-zero
idempotent a ∈ QH∗(M) by means of (2). Denote by ζ
′ the quasi-state on
M ′ associated to a⊗ T . Note that
Y = (H ′)−1(maxH ′) , where maxH ′ = 1 ,
while Theorem 5.1 and Lemma 6.2 imply that
ζ ′(H ′) = ζ(H) + 1 .
By Lemma 6.3 ζ ′(H ′) < 1 and so ζ(H) < 0. In view of Proposition 4.1, we
get a contradiction with the heaviness of X .
Lemma 6.2 also yields a simple proof of the following result which also follows
from Corollary 1.15:
Corollary 6.4. Any meridian of T2 is heavy (with respect to the fundamental
class [T ]).
Proof. In the notation as above identify T2 with T21 for R = 1. Since any
two meridians of T2 can be mapped into each other by a symplectic isotopy
and since such an isotopy preserves heaviness, it suffices to prove that the
meridian S := S+ = {r = 0} (see the proof of Lemma 6.2) is heavy.
Let H : T2 → R be a Hamiltonian and let us show that ζ(H) ≥ infSH ,
where ζ is defined using [T ]. Shifting H , if necessary, by a constant, we may
assume without loss of generality that infSH = 1. Pick f = f(r) : T
2 → R
as in the definition of Fα so that F1 = f ≤ H on T
2 (note that f equals 1 on
S). Then Lemma 6.2 yields
ζ(H) ≥ ζ(F1) = 1 = inf
7 Analyzing stable stems
Proof of Theorem 1.6: Assume that A is a Poisson-commutative subspace
of C∞(M), Φ : M → A∗ its moment map with the image ∆, and let X =
Φ−1(p) be a stable stem of A.
Take any functionH ∈ C∞(A∗) with H ≥ 0 andH(p) = 0. We claim that
ζ(Φ∗H) = 0. By an arbitrarily small C0-perturbation of H we can assume
that H = 0 in a small neighborhood, say U , of p. Choose an open covering
U0, U1, . . . , UN of ∆ so that U0 = U , and all Φ
−1(Ui) are stably displaceable
for i ≥ 1 (it exists by the definition of a stem). Let ρi : ∆ → R, i = 0, . . . , N ,
be a partition of unity subordinated to the covering {Ui}.
Take the two-torus T2R as in Section 6. Choose R > 0 large enough so that
Φ−1(Ui)× {r = const} is displaceable in M × T
R for all i ≥ 1. Choose now
a sufficiently fine covering Vj , j = 1, . . . , K, of the torus T
R by sufficiently
thin annuli {|r − rj | < δ} so that the sets Φ
−1(Ui) × Vj are displaceable in
M × T2R for all i ≥ 1 and all j. Let ̺j = ̺j(r), j = 1, . . . , K, be a partition
of unity subordinated to the covering {Vj}.
Denote by ζ ′ the partial quasi-state corresponding to a⊗T . Put F (r, θ) =
cos(2πr/R). Write
Φ∗H + F =
(Φ∗H + F ) · Φ∗ρi · ̺j =
Φ∗(Hρ0) + F · Φ
∗ρ0 +
(Φ∗H + F ) · Φ∗ρi · ̺j .
Note that Hρ0 = 0 and F · Φ
∗ρ0 ≤ 1. Applying partial quasi-additivity
and monotonicity we get that
ζ ′(Φ∗H + F ) = ζ ′(F · Φ∗ρ0) ≤ 1.
By Lemma 6.2 and the product formula (Theorem 5.1 above) we have
ζ ′(Φ∗H + F ) = ζ(Φ∗H) + 1 ≤ 1
and hence ζ(Φ∗H) ≤ 0. On the other hand, ζ(Φ∗H) ≥ 0 since H ≥ 0. Thus
ζ(Φ∗H) = 0 and the claim follows.
Further, given any function G on M with G ≥ 0 and G|X = 0, one can
find a function H on A∗ with H(p) = 0 so that G ≤ Φ∗H . By monotonicity
and the claim above
0 ≤ ζ(G) ≤ ζ(Φ∗H) = 0 ,
and hence ζ(G) = 0. Thus X is superheavy.
8 Monotone Lagrangian submanifolds
The main tool of proving (super)heaviness of monotone Lagrangian subman-
ifolds satisfying the Albers condition is the spectral estimate in Proposi-
tion 8.1(iii) below, which originated in the work by Albers [2]. Later on
Biran and Cornea pointed out a mistake in [2], and suggested a correction
together with a far reaching generalization in [15]. Let us mention that the
original Albers estimate was used in the first version of the present paper. We
thank Biran and Cornea for informing us about the mistake, explaining to
us their approach and helping us to correct a number of our results affected
by this mistake.
The main ingredient of Biran-Cornea techniques which is needed for our
purposes is the following result. Let (M,ω) be a closed monotone symplectic
manifolds with [ω] = κ·c1(M), κ > 0. WriteN for the minimal Chern number
of (M,ω). Let Ln ⊂M2n be a closed monotone Lagrangian submanifold with
the minimal Maslov number NL ≥ 2.
We shall treat slightly differently the cases when NL is even and odd. Let
us mention that for orientable L, NL is automatically even. Thus, due to
our convention, when NL is odd we work with the basic field F = Z2. Let
Γ = κN · Z be the group of periods of M . Recall that the quantum ring has
the form QH∗(M) = H∗(M ;F) ⊗F Λ, where the Novikov ring Λ is defined
as Λ = KΓ[q, q
−1] . Put Γ′ = (κN/2) · Z. Consider an extended Novikov ring
Λ′ := KΓ′ [q
2 , q−
2 ]. Define now QH ′∗(M) as QH∗(M) if NL is even, and as
H∗(M,Z2)⊗Z2 Λ
′ if NL is odd. In the latter case QH
∗(M) is an extension of
QH∗(M), and we shall consider without special mentioning QH∗(M), Λ, KΓ
as subrings of QH ′∗(M), Λ
′, KΓ′. The grading of QH
∗(M) is determined by
the condition deg q
2 = 1. As before, we shall use notation QH ′•(M), where
• = “even” when F = C and • = ∗ when F = Z2.
Note that the spectral invariants (and hence partial symplectic quasi-
states) are well-defined over the extended ring, and furthermore, their values
and properties, by tautological reasons, do not alter under such an extension
(cf. a discussion in [15], Section 5.4). Put w := sκNL/2qNL/2. Recall that j
stands for the natural morphism H•(L;F) → H•(M ;F).
Proposition 8.1. Assume that k > n+1−NL. If F = C assume in addition
that k is even. Then there exists a canonical homomorphism jq : Hk(L;F) →
QH ′k(M) with the following properties
8The letter “q” in jq stands for quantum.
(i) jq(x) = j(x) + w−1y, where y is a polynomial in w−1 with coefficients
in H•(M ;F);
(ii) jq([L]) = j([L]);
(iii) If jq(x) 6= 0 then c(jq(x), H) ≤ supLH for every H ∈ C
∞(M).
In particular, if S is an Albers element of L, we have jq(S) = j(S)+O(w−1) 6=
This proposition was proved by Biran and Cornea in [15] in the case
F = Z2: The map j
q is essentially the map iL appearing in Theorem A(iii)
in [15]. Proposition 8.1(i) above is a combination of Theorem A(iii) and
Proposition 4.5.1(i) in [15]. Our variable w corresponds to the variable t−1 in
[15], while our sNκqN corresponds to the variable s−1 in Section 2.1.2 of [15].
After such an adjustment of the notation, the formula w := sκNL/2qNL/2 above
can be extracted from Section 2.1.2 of [15]. For Proposition 8.1(ii) above
see Remark 5.3.2.a in [15]. Proposition 8.1(iii) above follows from Lemma
5.3.1(ii) in [15]. Finally, let us repeat the disclaimer made in Section 1.5: we
take for granted that Proposition 8.1 remains valid for the case F = C.
Let us pass to the proofs of our results on (super)-heaviness of monotone
Lagrangian submanifolds. We start with the following remark. Let S be an
Albers element of L. The Poincaré duality property of spectral invariants
(see Section 3.4 above) extends verbatim to the case of the ring QH ′(M)
with the following modification: When NL is odd, the pairing Π introduced
in Section 3.4 extends in the obvious way to a non-degenerate F -valued
pairing on QH ′•(M) which we still denote by Π. Applying Poincaré duality
and substituting H := −F into Proposition 8.1 (iii) above we get that for
every function F ∈ C∞(M)
c(T, F ) ≥ inf
F ∀T ∈ QH ′•(M) with Π(T, j
q(S)) 6= 0.
In particular, given a non-zero idempotent a ∈ QH ′•(M) and a class b ∈
QH ′•(M), so that Π(a∗b, j
q(S)) 6= 0, we get, using the normalization property
of spectral invariants, that
c(a, F ) + ν(b) ≥ c(a ∗ b, F ) ≥ inf
F ∀F ∈ C∞(M) . (27)
Therefore, applying (27) to kF for k ∈ N, dividing by k and passing to the
limit as k → +∞, we get that for the partial quasi-state ζ , defined by a,
ζ(F ) ≥ inf
F ∀F ∈ C∞(M),
meaning that L is heavy with respect to a.
Proof of Theorem 1.15: Let S be an Albers element of L. Let T ∈
H•(M ;F) be any singular homology class such that T ◦ j(S) 6= 0. Thus,
applying Proposition 8.1 (i) we see that Π([M ]∗T, jq(S)) = Π(T, jq(S)) 6= 0,
and hence inequality (27), applied to a = [M ], b = T , yields that L is heavy
with respect to [M ].
Let us pass to the proof of Theorem 1.25 on the effect of semi-simplicity
of the quantum homology. It readily follows from the next more general
statement. Let L1, . . . , Lm be Lagrangian submanifolds satisfying the Albers
condition. Let Si be any Albers element of Li. Denote by Zi = j
q(Si) ∈
QH ′•(M) its image under the inclusion morphism from Proposition 8.1 above.
Theorem 8.2. Given such L1, . . . , Lm and Z1, . . . , Zm, assume, in addition,
that QH2n(M) is semi-simple and the Lagrangian submanifolds L1, . . . , Lm
are pair-wise disjoint. Then the classes Z1, . . . , Zm are linearly independent
over KΓ′.
Proof. Arguing by contradiction, assume that
Z1 = α2Z2 + . . .+ αmZm (28)
for some α2, . . . , αm ∈ KΓ′ . Since QH2n(M) is semi-simple, it decomposes
into a direct sum of fields with unities e1, . . . , ed. Since the pairing Π (on
QH ′•(M ;F)) is non-degenerate, there exists T ∈ QH
•(M ;F) such that
Π(T, Z1) 6= 0. (29)
Let us write T as
T = [M ] ∗ T =
ei ∗ T. (30)
Equations (29), (30) imply that there exists l ∈ [1, d] such that
Π(el ∗ T, Z1) 6= 0 . (31)
Then (28) implies that there exists r ∈ [2, m] such that
Π(el ∗ T, αrZr) 6= 0.
Using (21) (for Π on QH ′•(M ;F)) we can rewrite the last equation as
Π(el ∗ αrT, Zr) 6= 0. (32)
Applying now formula (27) for S = Z1 ∈ H•(L1;F), a = el, b = T , and also
for S = Zr ∈ H•(Lr;F), a = el, b = αrT , we conclude that both L1 and Lr
are heavy with respect to el. Thus they are superheavy with respect to el,
because el is the unity in a field factor of QH2n(M) (see Section 1.6). Hence
they must intersect – in contradiction to the assumption of the theorem. This
finishes the proof of the first part of the theorem.
Proof of Theorem 1.25(a): Assume that L1, . . . , Lm are pair-wise disjoint
Lagrangian submanifolds satisfying the condition (a) from the formulation
of the theorem. Denote by Ni the minimal Maslov number of Li. Since
Ni > n + 1, the class of a point from H0(Li;F) is an Albers element for Li.
Let Zi ∈ QH
0(M) be its image under the Biran-Cornea inclusion morphism
associated to Li. Note that by Proposition 8.1(i) Zi = p + aiw
i , where
wi = s
κNi/2qNi/2, ai ∈ HNi(M ;F) and p ∈ H0(M ;F) is the homology class of
a point. Observe that degwi = Ni > n + 1, and hence the expression for Zi
cannot contain terms in w−1i of order two and higher, since HkNi(M ;F) = 0
for k ≥ 2.
Recall now that all Ni’s lie in some set Y of positive integers. Let W ⊂
QH ′0(M) be the span over KΓ′ of
H0(M ;F)⊕
s−κE/2q−E/2 ·HE(M ;F) .
Note that
dimKΓ′ W = βY (M) + 1 < m .
Thus the elements Zi, i = 1, . . . , m, are linearly dependent over KΓ′ . By
Theorem 8.2, QH2n(M) is not semi-simple.
Proof of Theorem 1.25(b): Assume that L1, . . . , Lm are pair-wise disjoint
homologically non-trivial Lagrangian submanifolds. By Proposition 8.1(ii)
jq([Li]) = j([Li]) for every i = 1, . . . , m. Since the classes j([Li]) are linearly
dependent, Theorem 8.2 implies that QH2n(M) is not semi-simple.
Proof of Theorem 1.18: Combining Proposition 8.1 (ii) and (iii) we get
that for any H ∈ C∞(M)
c(j([L]), H) ≤ sup
H ∀H ∈ C∞(M) .
By the hypothesis of the theorem, we can write j([L]) ∗ b = a for some b.
c(a,H) = c(j([L]) ∗ b,H) ≤ c(j([L]), H) + c(b, 0) .
c(a,H) ≤ sup
H + c(b, 0) .
Applying this inequality to E · H with E > 0, dividing by E and passing
to the limit as E → +∞ we get that ζ(H) ≤ supLH for all H . Thus L is
superheavy.
Remark 8.3. The results above admit the following generalizations in the
framework of the Biran-Cornea theory. The main object of this theory is the
quantum homology ring QH∗(L) of a monotone Lagrangian submanifold,
which is isomorphic to the Lagrangian Floer homology HF∗(L, L) of L up to
a shift of the grading.
(i) If QH∗(L) does not vanish then L is heavy (see Remark 1.2.9b in [15]).
In fact, it follows from [15] that if L satisfies the Albers condition,
QH∗(L) does not vanish.
(ii) The map jq of the Proposition 8.1 above is a footprint of the quan-
tum inclusion map iL : QH∗(L) → QH
∗(M) constructed in [15]. The
analogue of the action estimate in item (iii) of the proposition is ob-
tained in [15] for the classes iL(x) for elements x ∈ QH∗(L) of a certain
special form, yielding the following generalization of Theorem 1.18: for
these special classes x ∈ QH∗(L) the condition that the class iL(x)
does not vanish and divides a non-trivial idempotent a implies that L
is superheavy with respect to a. This enables, for instance, to general-
ize Example 1.19 on Lagrangian spheres in quadrics above to the case
when dimL is odd.
(iii) In [15] one can find another action estimate which comes from the
QH∗(M)-module structure on QH∗(L), which yields more results on
(super)heaviness of monotone Lagrangian submanifolds.
Proof of Proposition 1.4: The quantum homology QH2n(M) splits as an
algebra over K into a direct sum of two algebras one of which is a field. This
was proved by McDuff for the field F = C (see [39] and [24, Section 7]), but
the proof goes through for the case F = Z2 as well. Denote the unity of the
field by a. It is a non-zero idempotent in QH2n(M). As we already pointed
out in Remark 1.21, such an idempotent a defines a genuine symplectic quasi-
state and therefore the classes of heavy and superheavy sets with respect to
a coincide.
By Theorem 1.2, the Lagrangian torus L ⊂ M cannot be superheavy
with respect to a, since it can be displaced from itself by a symplectic (non-
Hamiltonian) isotopy. Indeed, take an obvious symplectic isotopy φt of T
that displaces L (a parallel shift) and compose it with a Hamiltonian isotopy
ψt so that for every t we have that ψt is constant on φt(L) and ψtφt is identity
on the ball where the blow up of T2n was performed. Clearly, the resulting
symplectic isotopy ψtφt extends to a symplectic isotopy of M that displaces
On the other hand, NL ≥ 2 because in this case NL = 2N , where N ≥ 1
is the minimal Chern number of M . Finally, note that L represents a non-
trivial homology class in Hn(M ;Z2). Therefore we can apply Theorem 1.15
and get that L is heavy with respect to [M ].
9 Rigidity of special fibers of Hamiltonian ac-
tions
In this section we prove Theorem 1.9. Denote the special fiber of Φ by
L := Φ−1(pspec).
Reduction to the case of T1-actions: First, we claim that it is enough
to prove the theorem for Hamiltonian T1-actions and the general case will
follow from it. Indeed, assume this is proved. The superheaviness of the
special fiber immediately yields that for any function H̄ : R → R
ζ(Φ∗H̄) = H̄(pspec), (33)
where Φ :M → R is the moment map of the T1-action.
Let us turn to the multi-dimensional situation and let Φ : M → Rk
be the normalized moment map of a Hamiltonian Tk-action on M . For a
v ∈ Rk denote by Φv(x) = 〈v,Φ(x)〉, where 〈·, ·〉 is the standard Euclidean
inner product on Rk. Note that if v ∈ Zk the function Φv is the normalized
moment map of a Hamiltonian circle action and its special value is 〈v, pspec〉.
Thus by (33)
K) = K(〈v, pspec〉) ∀K ∈ C
∞(R) . (34)
By homogeneity of ζ , equality (34) holds for all v ∈ Qk, and by continuity
for all v ∈ Rk.
Observe that for each pair of smooth functions P,Q ∈ C∞(R) and for each
pair of vectors v,w ∈ Rk the functions Φ∗
P and Φ∗
Q Poisson-commute on
M . Thus the triangle inequality for the spectral numbers (see Section 3.4)
yields
P + Φ∗
Q) ≤ ζ(Φ∗
P ) + ζ(Φ∗
Q) . (35)
Since M is compact, it suffices to assume that the function H̄ ∈ C∞(Rk) on
Rk is compactly supported. By the inverse Fourier transform we can write
H̄(p) =
sin〈v, p〉 · F (v) + cos〈v, p〉 ·G(v)
for some rapidly (say, faster than (|p| + 1)−N for any N ∈ N) decaying
functions F and G on Rk. For every v ∈ Rk define a function Kv ∈ C
Kv(s) := sin s · F (v) + cos s ·G(v) .
Observe that
Φ∗H̄ =
Kv dv .
Denote by B(R) the Euclidean ball of radius R in Rk with the center at the
origin. Put
H̄R(p) =
Kv(〈v, p〉) dv, p ∈ R
Since the functions F and G are rapidly decaying, we get that
||H̄R − H̄||C0(Rk) → 0 as R → ∞ . (36)
We claim that for every R
ζ(Φ∗H̄R) ≤ H̄R(pspec) . (37)
Indeed, for ǫ > 0 introduce the integral sum
H̄R,ε(p) =
v∈ ε·Zk∩B(R)
εk ·Kv(〈v, p〉) .
Φ∗H̄R,ε =
v∈ ε·Zk∩B(R)
εk · Φ∗
Applying repeatedly (35) and (34) we get that
ζ(Φ∗H̄R,ε) ≤ H̄R,ε(pspec) .
Note now that for fixed R the family H̄R,ǫ converges to H̄R as ε → 0 in
the uniform norm on C0(Rk). Using that ζ is Lipschitz with respect to the
uniform norm on C0(M) we readily get the inequality (37).
Combining the fact that ζ is Lipschitz with (36) and (37) we get that
ζ(Φ∗H̄) = lim
ζ(Φ∗H̄R) ≤ lim
H̄R(pspec) = H̄(pspec) .
Now, assume that H̄ ≥ 0 and H̄(pspec) = 0. We just have proved that
ζ(Φ∗H̄) ≤ 0, and hence ζ(H) = 0, which immediately yields the desired su-
perheaviness of the special fiber. This completes the reduction of the general
case to the 1-dimensional case.
From now on we will consider only the case of an effective Hamil-
tonian T1-action on M with a moment map Φ :M → R. Its moment
polytope ∆ is a closed interval in R and pspec = −I(Φ) ∈ R.
Reduction to the case of a strictly convex function: We claim
that it is enough to show the following proposition:
Proposition 9.1. Assume H̄ : R → R is a strictly convex smooth function
reaching its minimum at pspec. Set H := Φ
∗H̄. Then ζ(H) = H̄(pspec).
Postponing the proof of the proposition for a moment let us show that it
implies the theorem. Indeed, let F : M → R be a Hamiltonian on M . In
order to show the superheaviness of L = Φ−1(pspec) we need to show that
ζ(F ) ≤ supL F . Pick a very steep strictly convex function H̄ : R → R with
the minimum value supL F reached at pspec and such that Φ
∗H̄ =: H ≥ F
everywhere on M . Then using Proposition 9.1 and the monotonicity of ζ we
ζ(F ) ≤ ζ(H) = H̄(pspec) = sup
yielding the claim.
Preparations for the proof of Proposition 9.1: Given a (time-
dependent, not necessarily regular) Hamiltonian G, we associate to every
pair [γ, u] ∈ P̃G a number
DG([γ, u]) := AG([γ, u])−
· CZG([γ, u]).
(Recall that we defined the Conley-Zehnder index for all Hamiltonians and
not only the regular ones – see Section 3.3). The number DG([γ, u]) is in-
variant under a change of the spanning disc u – an addition of a sphere
jS ∈ HS2 (M) to the disc u changes both AG([γ, u]) and κ/2 · CZG([γ, u]) by
the same number. Thus we can write DG([γ, u]) = DG(γ).
Given [γ, u] ∈ P̃G and l ∈ N define γ
(l) and u(l) as the compositions
of γ and u with the map z → zl on the unit disc D2 ⊂ C (here z is a
complex coordinate on C). Denote by t 7→ gt the time-t flow of G and by
G(l) :M × R → R the Hamiltonian whose time-t flow is t 7→ (gt)
l and which
is defined by
G(l) := G♯ . . . ♯G (l times),
where G♯K(x, t) := G(x, t) +K(g−1t x, t) for any K :M × R → R.
Proposition 9.2. There exists a constant C > 0, depending only on n, with
the following property. Given a 1-periodic orbit γ ∈ PG of the flow t 7→ gt
generated by G, assume that γ(l) is a 1-periodic orbit of the flow t 7→ glt
generated by G(l), and therefore for any u such that [γ, u] ∈ P̃G we have
[γ(l), u(l)] ∈ P̃G(l). Then
|DG(l)([γ
(l), u(l)])− lDG([γ, u])| ≤ l · C.
Proof. The action term in DG gets multiplied by l as we pass from G to G
As for the Conley-Zehnder term, the quasi-morphism property of the Conley-
Zehnder index (see Proposition 3.5) implies that there exists a constant C > 0
(depending only on n) such that
|lCZG[γ, u]− CZG(l)([γ
(l), u(l)])| ≤ C.
This immediately proves the proposition.
Proposition 9.3. Let G :M × [0, 1] → R be a Hamiltonian as above. Then
one can choose ǫ > 0, depending on G, and a constant Cn > 0, depending
only on n = dimM/2, so that any function F : M × [0, 1] → R which is ǫ-
close to G in a C∞-metric on C∞(M×[0, 1]) satisfies the following condition:
for every γ0 ∈ PF there exists γ ∈ PG such that the difference between DF (γ0)
and DG(γ) is bounded by Cn.
Proof. Denote the flow of G by gt (as before) and the flow of F by ft. We
will view time-1 periodic trajectories of these flows both as maps of [0, 1] to
M having the same value at 0 and 1 and as maps from S1 to M .
First, consider the fibration D2×M →M and, slightly abusing notation,
denote the natural pullback of ω again by ω. Second, look at the fibration
pr : D2 ×M → D2. Denote by V ert the vertical bundle over D2 ×M formed
by the tangent spaces to the fibers of pr. For each loop σ : S1 →M define by
σ̂ : S1 → D2 ×M the map σ̂(t) := (t, γ(t)). The bundles σ∗TM and σ̂∗V ert
over S1 coincide. Similarly for each w : D2 →M denote by ŵ : D2 → D2×M
the map ŵ(z) := (z, w(z)).
There exists δ > 0, depending on G, such that for each γ ∈ PG a tubular
δ-neighborhood of the image of γ̂ in S1 ×M ⊂ D2 ×M , denoted by Ubγ, has
the following properties:
• there exists a 1-form λ on Ubγ satisfying dλ = ω;
• V ert admits a trivialization over Ubγ .
Given an ǫ > 0, we can choose F sufficiently C∞-close to G so that the
paths t 7→ ft and t 7→ gt in Ham(M) are arbitrarily C
∞-close and therefore
• for every x ∈ Fix (F ) there exists y ∈ Fix (G) which is ǫ-close to x
(think of the fixed points as points of intersection of the graph of a
diffeomorphism with the diagonal);
• the C∞-distance between the maps γ0 : t 7→ ft(x) and γ : t 7→ gt(y)
from [0, 1] to M is bounded by ǫ and the image of γ̂0 lies in Ubγ.
Pick a map u0 : D
2 → M , u|∂D2 = γ0. Since γ0 and γ are C
∞-close one
can enlarge D2 to a bigger disc D21 ⊃ D
2 and find a smooth map u : D21 →M
so that
• u|∂D21
= u0;
• u(D21 \ D
2) ⊂ Ubγ.
Rescaling D21 we may assume without loss of generality that [γ, u] ∈ PG.
Trivialize the vector bundles γ∗0TM and γ
∗TM so that the trivializations
extend to a trivialization of u∗TM over D21 (and hence of u
0TM over D
Using the trivializations we can identify the paths t 7→ dγ0(0)ft and t 7→ dγ(0)gt
with some identity-based paths of symplectic matrices A(t), B(t). Fixing a
small ǫ as above, we can also assume that F is chosen so C∞-close to G that,
in addition to all of the above, the C∞-distance between the paths t 7→ A(t)
and t 7→ B(t) in Sp (2n) is bounded by ǫ (for instance, make sure first that
the matrix paths obtained by writing the paths t 7→ dγ0(0)ft and t 7→ dγ(0)gt
using some trivialization of V ert over Ubγ are close enough – then the matrix
paths t 7→ A(t) and t 7→ B(t) will also be close enough).
We claim that by choosing ǫ sufficiently small in the construction above we
can bound the difference between DF ([γ0, u0]) and DG([γ, u]) by a quantity
depending only on dimM .
Indeed, the difference |
F (γ0(t), t)dt −
G(γ(t))dt| is bounded by a
quantity depending only on some universal constants and ǫ, because γ0 is
ǫ-close to γ and F is ǫ-close to G with respect to the C∞-metrics. It can be
made arbitrarily small by choosing a sufficiently small ǫ. The difference
u∗0ω −
u∗ω| = |
û∗0ω −
û∗ω|
is bounded by the difference |
γ̂∗0λ −
γ̂∗λ|. Since, γ0 and γ are ǫ-close
in the C∞-metric the later difference can be made less than 1 if we choose
a sufficiently small ǫ. Thus we have shown that by choosing a sufficiently
small ǫ we can bound |AF ([γ0, u0])−AG([γ, u])| by 1.
Now, as far as the Conley-Zehnder indices are concerned, our choice
of the trivializations means that the difference between CZF ([γ0, u0]) and
CZG([γ, u]) is just the difference between the Conley-Zehnder indices for the
matrix paths t 7→ A(t) and t 7→ B(t). But the latter paths in Sp (2n) are
ǫ-close in the C∞-sense, hence represent close elements of S̃p (2n) and if ǫ
was chosen sufficiently small, then, as we mentioned in Section 3.3, their
Conley-Zehnder indices differ at most by a constant depending only on n.
This finishes the proof of the claim and the proposition.
Plan of the proof of Proposition 9.1: We assume now that H̄ is
a fixed strictly convex function on R. Our calculations will feature E as a
large parameter. For quantities α, β depending on E we will write α � β if
α ≤ β+const holds for large enough E, where const depends only on (M,ω),
Φ and H̄ , and in particular does not depend on E. We will write α ≈ β if
α � β and β � α. Using this language the proposition can be restated as
c(a, EH) ≈ EH̄(pspec). (38)
In general, 1-periodic orbits of the flow of EH are not isolated and there-
fore the Hamiltonian is not regular. Let F be a regular (time-periodic)
perturbation of EH .
By the spectrality axiom, the spectral number c(a, F ) for a ∈ QH2n(M)
equals AF ([γ0, u0]) for some pair [γ0, u0] ∈ P̃F with CZF ([γ0, u0]) = 2n. Thus
c(a, F ) ≈ DF (γ0). Combining this with Proposition 9.3 we get that for some
γ ∈ PEH
EH̄(pspec) � c(a, EH) ≈ c(a, F ) ≈ DF (γ0) ≈ DEH(γ) . (39)
Thus it would be enough to show that
DEH(γ) � EH̄(pspec) for all γ ∈ PEH , (40)
which together with (39) would imply (38).
Inequality (40) will be proved in the following way. Note that each γ ∈
PEH lies in Φ
−1(p) for some p ∈ ∆. We will show that
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p). (41)
Note that (41) implies (40). Indeed, since H̄ is strictly convex and reaches
its minimum at pspec, it follows from (41) that
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p) ≤ EH̄(pspec),
which is true for any γ ∈ PEH thus yielding (40).
Proof of (41): Let the T1-action on M be given by a loop of sym-
plectomorphisms {φt}, t ∈ R, φt = φt+1. The flow of EH has the form
htx = φEH̄′(Φ(x))tx.
We view γ as a map γ : [0, 1] → M satisfying γ(0) = γ(1). Denote
x := γ(0). The curve γ lies in Φ−1(p).
Denote N := γ([0, 1]). This is the T1-orbit of x and it is either a point or
a circle.
In the first case γ is a constant trajectory concentrated at a fixed point
N ∈M of the action. Using this constant curve γ together with the constant
disc u spanning for the definitions of I(Φ) and DEH(γ) one gets
pspec − p = mΦ(γ, u) · κ/2,
DEH(γ) = EH̄(p)− κ/2 · CZEH([γ, u]).
Thus proving (41) reduces in this case to proving
−CZEH([γ, u]) ≈ EH̄
′(p) ·mΦ(γ, u).
Let us fix a symplectic basis of TNM and view each differential dNφt as a
symplectic matrix A(t), so that {A(t)} is an identity-based loop in Sp (2n).
−CZEH([γ, u]) ≈ CZmatr({A(EH̄
′(p)t)}),
while
EH̄ ′(p) ·mΦ(γ, u) ≈ EH̄
′(p)Maslov({A(t)}).
Thus we need to prove
CZmatr({A(EH̄
′(p)t)}) ≈ EH̄ ′(p)Maslov({A(t)}),
which follows easily from the definitions of the Conley-Zehnder index and
the Maslov class.
Thus from now on we will assume that N is a circle. Take any point
x ∈ N . The stabilizer of x under the T1-action is a finite cyclic group of
order k ∈ N. Thus the orbit of the T1-action turns k times along N . Since γ
is a non-constant closed orbit of the Hamiltonian flow generated by EΦ∗H̄ ,
it turns r times along N with r ∈ Z \ {0}. This implies that EH̄ ′(p) = r/k.
We claim that without loss of generality we may assume that l := r/k is an
integer.
Indeed, we can always pass to γ(k) ∈ PkEH , so that (kEH̄)
′(p) ∈ Z, and
if we can prove the proposition for γ(k), then
DkEH(γ
(k)) ≈ kEH̄(p) + kEH̄ ′(p)(pspec − p).
Applying Proposition 9.2 we get
kDEH(γ) ≈ kEH̄(p) + kEH̄
′(p)(pspec − p) + k · const ,
and hence
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p),
proving the claim for the original γ.
From now on we assume that l := EH̄ ′(p) ∈ Z\{0} and that [γ, u] ∈ P̃lΦ.
Consider the Hamiltonian vector field X := sgradΦ at a point x ∈ N . Since
N is a non-constant orbit we get X 6= 0. Then V = Tx(Φ
−1(p)) is the skew-
orthogonal complement to X . Choose a T1-invariant ω-compatible almost
complex structure J in a neighborhood of N . Together ω and J define a T1-
invariant Riemannian metric g. Decompose the tangent bundle TM along
N as follows. Put Z = Span(JX,X) and set W to be the g-orthogonal
complement to X in V . Thus we have a T1-invariant decomposition
TxM = W ⊕ Z , x ∈ N . (42)
Furthermore, W and Z carry canonical symplectic forms. Thus W and Z
define symplectic (and hence trivial) subbundles of TM over N . They induce
trivial subbundles of the bundle γ∗TM over S1.
We calculate
dht(x)ξ = dφEH′(Φ(x))t(x)ξ + EH
′′(Φ(x)) · dΦ(ξ) ·X . (43)
We consider two trivializations of the bundle γ∗TM over S1. The first trivi-
alization is defined by means of sections invariant under the T1-action. The
second one is chosen in such a way that it extends to a trivialization of u∗TM
over D2. Using these trivializations we can identify dht(x), respectively, with
two identity-based paths {Ct}, {C
t} of symplectic matrices. The decompo-
sition (42) induces a split
Ct = 1⊕ Bt .
We claim that |CZmatr({Bt})| is bounded by a constant independent of E.
Indeed, observe that in the basis (X, JX) of Z
1 b12(t)
Denote by L the line spanned by X = (1, 0). Perturb {Bt} to a path {B
RδtBt}, where Rt is the rotation by angle t, and δ > 0 is small enough.
Observe thatB′(t)L∩L = {0} for t > 0. It follows readily from the definitions
that |CZmatr(B
t)| and |CZmatr(Rδt)| do not exceed 2. Thus by the quasi-
morphism property of the Conley-Zehnder index (see Proposition 3.5) we
have that |CZmatr({Bt})| is bounded by a constant independent of E, which
yields the claim. Therefore
CZmatr ({Ct}) ≈ 0 .
On the other hand, by formula (18)
CZmatr ({C
t}) = CZmatr ({Ct}) +mlΦ([γ, u]) .
CZEH([γ, u]) := n− CZmatr ({C
t}) ≈ −mlΦ([γ, u]). (44)
Since the periodic trajectory γ lies inside Φ−1(p), we get
AEH([γ, u]) =
EH(γ(t))dt−
u∗ω = EH̄(p)−
u∗ω. (45)
Using (45) and (44) the precise equality
DEH([γ, u]) = AEH([γ, u])−
· CZEH([γ, u])
can be turned into an asymptotic inequality
DEH([γ, u]) ≈ EH̄(p)−
u∗ω +
mlΦ([γ, u]). (46)
Since the periodic trajectory γ lies inside Φ−1(p), we have
AlΦ([γ, u]) =
lΦ(γ(t))dt−
u∗ω = lp−
u∗ω. (47)
Adding and subtracting lp from the right-hand side of (46) and using (47)
we get
DEH(γ) = DEH([γ, u]) ≈
EH̄(p)− lp)
mlΦ([γ, u])
EH̄(p)− lp
AlΦ([γ, u])+
mlΦ([γ, u])
EH̄(p)− lp
− I(lΦ) =
= EH̄(p) + l(−I(Φ)− p) = EH̄(p) + l(pspec − p) .
Recalling that l = EH ′(p), we finally obtain that
DEH(γ) = EH̄(p) + EH
′(p)(pspec − p),
which is precisely the equation (41) that we wanted to get. This finishes the
proof of Proposition 9.1 and Theorem 1.9.
9.1 Calabi and mixed action-Maslov
Proof of Theorem 1.13.
Assume H :M × [0, 1] → R is a normalized Hamiltonian which generates
a loop in Ham(M) representing a class α ∈ π1(Ham(M)) ⊂ H̃am (M). Then
H(l) is also normalized and generates a loop representing αl. Let us compute
µ(α) = −vol (M) · liml→+∞ c(a,H
(l))/l.
Arguing as in the proof of (39) we get that there exists a constant C >
0 such that for each l ∈ N there exists γ ∈ PH(l) for which |c(a,H
(l)) −
DH(l)(γ)| ≤ C. But, as it follows from the definitions and from the fact that
I is a homomorphism, DH(l)(γ) does not depend on γ and equals −I(α
−lI(α). This immediately implies that µ(α) = vol (M) · I(α).
Acknowledgements. The origins of this paper lie in our joint work with
P.Biran on the paper [10] – we thank him for fruitful collaboration at an
early stage of this project, as well as for his crucial help with Example 1.17
on Lagrangian spheres in projective hypersurfaces. We also thank him and
O.Cornea for pointing out to us a mistake in the original version of this paper
and helping us with the correction (see Section 8). We thank F. Zapolsky
for his help with the “exotic” monotone Lagrangian torus in S2 × S2 dis-
cussed in Example 1.20. We thank C. Woodward for pointing out to us the
link between the special point in the moment polytope of a symplectic toric
manifold and the Futaki invariant, and E. Shelukhin for useful discussions
on this issue. We are also grateful to V.L. Ginzburg, Y. Karshon, Y. Long,
D. McDuff, M. Pinsonnault, D. Salamon and M. Sodin for useful discussions
and communications. We thank K. Fukaya, H. Ohta and K. Ono, the orga-
nizers of the Conference on Symplectic Topology in Kyoto (February 2006),
M. Harada, Y. Karshon, M. Masuda and T. Panov, the organizers of the Con-
ference on Toric Topology in Osaka (May 2006), O. Cornea, V.L. Ginzburg,
E. Kerman and F. Lalonde, the organizers of the Workshop on Floer theory
(Banff, 2007), and A. Fathi, Y.-G. Oh and C. Viterbo, the organizers of the
AMS Summer Conference on Symplectic Topology and Measure-Preserving
Dynamical Systems (Snowbird, July 2007), for giving us an opportunity to
present a preliminary version of this work and for the superb job they did
in organizing these conferences. Finally, we thank an anonymous referee for
helpful comments and corrections.
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Michael Entov Leonid Polterovich
Department of Mathematics School of Mathematical Sciences
Technion Tel Aviv University
Haifa 32000, Israel Tel Aviv 69978, Israel
entov@math.technion.ac.il polterov@post.tau.ac.il
http://arxiv.org/abs/0709.1127
	Introduction and main results
	Many facets of displaceability
	Preliminaries on quantum homology
	An hierarchy of rigid subsets within Floer theory
	Hamiltonian torus actions
	Super(heavy) monotone Lagrangian submanifolds
	An effect of semi-simplicity
	Discussion and open questions
	Strong displaceability beyond Floer theory? 
	Heavy fibers of Poisson-commutative subspaces
	Detecting stable displaceability
	Preliminaries on Hamiltonian Floer theory
	Valuation on QH* (M)
	Hamiltonian Floer theory
	Conley-Zehnder and Maslov indices
	Spectral numbers
	Partial symplectic quasi-states
	Basic properties of (super)heavy sets
	Products of (super)heavy sets 
	Product formula for spectral invariants
	Decorated Z2-graded complexes
	Reduced Floer and Quantum homology
	Proof of Theorem 5.1
	Proof of algebraic Theorem 5.2 
	Stable non-displaceability of heavy sets
	Analyzing stable stems
	Monotone Lagrangian submanifolds 
	Rigidity of special fibers of Hamiltonian actions
	Calabi and mixed action-Maslov

  We show that there is an hierarchy of intersection rigidity properties of
sets in a closed symplectic manifold: some sets cannot be displaced by
symplectomorphisms from more sets than the others. We also find new examples of
rigidity of intersections involving, in particular, specific fibers of moment
maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following
the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly
singular, sets defined in terms of Poisson-commutative subalgebras of smooth
functions. In addition, we get some geometric obstructions to semi-simplicity
of the quantum homology of symplectic manifolds. The proofs are based on the
Floer-theoretical machinery of partial symplectic quasi-states.

Introduction and main results 3
1.1 Many facets of displaceability . . . . . . . . . . . . . . . . . . 3
1.2 Preliminaries on quantum homology . . . . . . . . . . . . . . . 8
1.3 An hierarchy of rigid subsets within Floer theory . . . . . . . 10
1.4 Hamiltonian torus actions . . . . . . . . . . . . . . . . . . . . 14
1.5 Super(heavy) monotone Lagrangian submanifolds . . . . . . . 19
1.6 An effect of semi-simplicity . . . . . . . . . . . . . . . . . . . . 23
1.7 Discussion and open questions . . . . . . . . . . . . . . . . . . 27
1.7.1 Strong displaceability beyond Floer theory? . . . . . . 27
1.7.2 Heavy fibers of Poisson-commutative subspaces . . . . 28
2 Detecting stable displaceability 32
3 Preliminaries on Hamiltonian Floer theory 33
3.1 Valuation on QH∗(M) . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Hamiltonian Floer theory . . . . . . . . . . . . . . . . . . . . 34
3.3 Conley-Zehnder and Maslov indices . . . . . . . . . . . . . . . 36
3.4 Spectral numbers . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Partial symplectic quasi-states . . . . . . . . . . . . . . . . . . 44
4 Basic properties of (super)heavy sets 45
5 Products of (super)heavy sets 48
5.1 Product formula for spectral invariants . . . . . . . . . . . . . 48
5.2 Decorated Z2-graded complexes . . . . . . . . . . . . . . . . . 49
5.3 Reduced Floer and Quantum homology . . . . . . . . . . . . . 50
5.4 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Proof of algebraic Theorem 5.2 . . . . . . . . . . . . . . . . . 52
6 Stable non-displaceability of heavy sets 57
7 Analyzing stable stems 59
8 Monotone Lagrangian submanifolds 61
9 Rigidity of special fibers of Hamiltonian actions 66
9.1 Calabi and mixed action-Maslov . . . . . . . . . . . . . . . . . 76
1 Introduction and main results
1.1 Many facets of displaceability
A well-studied and easy to visualize rigidity property of subsets of a symplec-
tic manifold (M,ω) is the rigidity of intersections: a subset X ⊂ M cannot
be displaced from the closure of a subset Y ⊂ M by a compactly supported
Hamiltonian isotopy:
φ(X) ∩ Y 6= ∅ ∀φ ∈ Ham(M) .
We say in such a case that X cannot be displaced from Y . If X cannot be
displaced from itself we call it non-displaceable. These properties become
especially interesting and purely symplectic when X can be displaced from
itself or from Y by a (compactly supported) smooth isotopy.
One of the main themes of the present paper is that “some non-displace-
able sets are more rigid than others.” To explain this, we need the following
ramifications of the notion of a non-displaceable set:
Strong non-displaceability: A subset X ⊂ M is called strongly non-
displaceable if one cannot displace it by any (not necessarily Hamiltonian)
symplectomorphism of (M,ω).
Stable non-displaceability: Consider T ∗S1 = R × S1 with the coordi-
nates (r, θ) and the symplectic form dr ∧ dθ. We say that X ⊂ M is stably
non-displaceable if X × {r = 0} is non-displaceable in M × T ∗S1 equipped
with the split symplectic form ω̄ = ω ⊕ (dr ∧ dθ). Let us mention that de-
tecting stably non-displaceable subsets is useful for studying geometry and
dynamics of Hamiltonian flows (see for instance [50] for their role in Hofer’s
geometry and [51] for their appearance in the context of kick stability in
Hamiltonian dynamics).
Formally speaking, the properties of strong and stable non-displaceability
are mutually independent and both are strictly stronger than displaceability.
In the present paper we refine the machinery of partial symplectic quasi-
states introduced in [23] and get new examples of stably non-displaceable
sets, including certain fibers of moment maps of Hamiltonian torus actions
as well as monotone Lagrangian submanifolds discussed by Albers [2] and
Biran-Cornea [15]. Further, we address the following question: given the class
of stably non-displaceable sets, can one distinguish those of them which are
also strongly non-displaceable by means of the Floer theory? Or, other way
around, what are the Floer-homological features of stably non-displaceable
but strongly displaceable sets? Toy examples are given by the equator of the
symplectic two-sphere and by the meridian on a symplectic two-torus. Both
are stably non-displaceable since their Lagrangian Floer homologies are non-
trivial. On the other hand, the equator is strongly non-displaceable, while
the meridian is strongly displaceable by a non-Hamiltonian shift. Later on we
shall explain the difference between these two examples from the viewpoint
of Hamiltonian Floer homology and present various generalizations.
The question on Floer-homological characterization of (strongly) non-displa-
ceable but stably displaceable sets is totally open, see Section 1.7.1 below for
an example involving Gromov’s packing theorem and discussion.
Leaving Floer-theoretical considerations for the next section, let us outline
(in parts, informally) the general scheme of our results: Given a symplectic
manifold (M,ω), we shall define (in the language of the Floer theory) two
collections of closed subsets of M , heavy subsets and superheavy subsets.
Every superheavy subset is heavy, but, in general, not vice versa. Formally
speaking, the hierarchy heavy-superheavy depends in a delicate way on the
choice of an idempotent in the quantum homology ring ofM . This and other
nuances will be ignored in this outline. The key properties of these collections
are as follows (see Theorems 1.2 and 1.5 below):
Invariance: Both collections are invariant under the group of all symplec-
tomorphisms of M .
Stable non-displaceability: Every heavy subset is stably non-displace-
able.
Intersections: Every superheavy subset intersects every heavy subset. In
particular, superheavy subsets are strongly non-displaceable. In contrast to
this, heavy subsets can be mutually disjoint and strongly displaceable.
Products: Product of any two (super)heavy subsets is (super)heavy.
What is inside the collections? The collections of heavy and superheavy
sets include the following examples:
Stable stems: Let A ⊂ C∞(M) be a finite-dimensional Poisson-commuta-
tive subspace (i.e. any two functions from A commute with respect to the
Poisson brackets). Let Φ :M → A∗ be the moment map: 〈Φ(x), F 〉 = F (x).
A non-empty fiber Φ−1(p), p ∈ A∗, is called a stem of A (see [23]) if all
non-empty fibers Φ−1(q) with q 6= p are displaceable and a stable stem if
they are stably displaceable. If a subset of M is a (stable) stem of a finite-
dimensional Poisson-commutative subspace of C∞(M), it will be called just
a (stable) stem. Clearly, any stem is a stable stem. The collection of
superheavy subsets includes all stable stems (see Theorem 1.6 below).
One readily shows that a direct product of stable stems is a stable stem and
that the image of a stable stem under any symplectomorphism is again a
stable stem.
The following example of a stable stem is borrowed (with a minor mod-
ification) from [23]: Let X ⊂ M be a closed subset whose complement is
a finite disjoint union of stably displaceable sets. Then X is a stable stem.
For instance, the codimension-1 skeleton of a sufficiently fine triangulation of
any closed symplectic manifold is a stable stem. Another example is given by
the equator of S2: it divides the sphere into two displaceable open discs and
hence is a stable stem. By taking products, one can get more sophisticated
examples of stable stems. Already the product of equators of the two-spheres
gives rise to a Lagrangian Clifford torus in S2× . . .×S2. To prove its rigidity
properties (such as stable non-displaceability) one has to use non-trivial sym-
plectic tools such as Lagrangian Floer homology, see e.g. [44]. Products of
the 1-skeletons of fine triangulations of the two-spheres can be considered as
singular Lagrangian submanifolds, an object which is currently out of reach
of the Lagrangian Floer theory.
Another example of stable stems comes from Hamiltonian torus actions.
Consider an effective Hamiltonian action ϕ : Tk → Ham(M) with the mo-
ment map Φ = (Φ1, . . . ,Φk) : M → R
k. Assume that Φi is a normalized
Hamiltonian, that is
Φi = 0 for all i = 1, . . . , k. A torus action is called
compressible if the image of the homomorphism ϕ♯ : π1(T
k) → π1(Ham(M)),
induced by the action ϕ, is a finite group. One can show that for compressible
actions the fiber Φ−1(0) is a stable stem (see Theorem 1.7 below).
Special fibers of Hamiltonian torus actions: Consider an effective
Hamiltonian torus action ϕ on a spherically monotone symplectic manifold.
Let I : π1(Ham(M)) → R be the mixed action-Maslov homomorphism intro-
duced in [49]. Since the target space Rk of the moment map Φ is naturally
identified with Hom(π1(T
k),R), the pull back pspec := −ϕ
♯I of the mixed
action-Maslov homomorphism with the reversed sign can be considered as a
point of Rk. The preimage Φ−1(pspec) is called the special fiber of the action.
We shall see below that the special fiber is always non-empty. For monotone
symplectic toric manifolds (that is when 2k = dimM) the special fiber is a
monotone Lagrangian torus. Note that when the action is compressible we
have pspec = 0 and therefore the special fiber is a stable stem according to the
previous example. It is unknown whether the latter property persists for gen-
eral non-compressible actions. Thus in what follows we treat stable stems
and special fibers as separate examples. The collection of superheavy
subsets includes all special fibers (see Theorem 1.9 below).
For instance, consider CP 2 and the Lagrangian Clifford torus in it (i.e.
the torus {[z0 : z1 : z2] ∈ CP
2 | |z0| = |z1| = |z2|}). Take the standard
Hamiltonian T2-action on CP 2 preserving the Clifford torus. It has three
global fixed points away from the Clifford torus. Make an equivariant sym-
plectic blow-up, M , of CP 2 at k of these fixed points, 0 ≤ k ≤ 3, so that
the obtained symplectic manifold is spherically monotone. The torus action
lifts to a Hamiltonian action on M . One can show that its special fiber is
the proper transform of the Clifford torus.
Monotone Lagrangian submanifolds: Let (M2n, ω) be a spherically
monotone symplectic manifold, and let L ⊂ M be a closed monotone La-
grangian submanifold with the minimal Maslov number NL ≥ 2. We say
that L satisfies the Albers condition [2] if the image of the natural morphism
H∗(L;Z2) → H∗(M ;Z2) contains a non-zero element S with
deg S > dimL+ 1−NL .
The collection of heavy sets includes all closed monotone Lagran-
gian submanifolds satisfying the Albers condition (see Theorem 1.15
below).
Specific examples include the meridian on T2, RP n ⊂ CP n and all La-
grangian spheres in complex projective hypersurfaces of degree d in CP n+1
with n > 2d − 3. In the case when the fundamental class [L] of L divides
a non-trivial idempotent in the quantum homology algebra of M , L is, in
fact, superheavy (see Theorem 1.18 below). For instance, this is the case
for RP n ⊂ CP n. Furthermore, a version of superheaviness holds for any
Lagrangian sphere in the complex quadric of even (complex) dimension.
However, there exist examples of heavy, but not superheavy, Lagrangian
submanifolds: For instance, the meridian of the 2-torus is strongly displa-
ceable by a (non-Hamiltonian!) shift and hence is not superheavy. Another
example of heavy but not superheavy Lagrangian submanifold is the sphere
arising as the real part of the Fermat hypersurface
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
with even d ≥ 4 and n > 2d− 3. We refer to Section 1.5 for more details on
(super)heavy monotone Lagrangian submanifolds.
Motivation: Our motivation for the selection of examples appearing in the
list above is as follows. Stable stems provide a playground for studying
symplectic rigidity of singular subsets. In particular, no visible analogue of
the conventional Lagrangian Floer homology technique is applicable to them.
Detecting (stable) non-displaceability of Lagrangian submanifolds via La-
grangian Floer homology is one of the central themes of symplectic topology.
In contrast to this, detecting strong non-displaceabilty has at the moment the
status of art rather than science. That’s why we were intrigued by Albers’
observation that monotone Lagrangian submanifolds satisfying his condition
are in some situations strongly non-displaceable. In the present work we tried
to digest Albers’ results [2] and look at them from the viewpoint of theory
of partial symplectic quasi-states developed in [23]. In addition, our result
on superheaviness of the Lagrangian anti-diagonal in S2 × S2 allows us to
detect an “exotic” monotone Lagrangian torus in this symplectic manifold:
this torus does not intersect the anti-diagonal, and hence is not heavy in
contrast to the standard Clifford torus, see Example 1.20 below.
In [23] we proved a theorem which roughly speaking states that every
(singular) coisotropic foliation has at least one non-displaceable fiber. How-
ever, our proof is non-constructive and does not tell us which specific fibers
are non-displaceable. The notion of the special fiber arose as an attempt to
solve this problem for Hamiltonian circle actions.
Let us mention also that the product property enables us to produce even
more examples of (super)heavy subsets by taking products of the subsets
appearing in the list.
A few comments on the methods involved into our study of heavy and su-
perheavy subsets are in order. These collections are defined in terms of
partial symplectic quasi-states which were introduced in [23]. These are cer-
tain real-valued functionals on C∞(M) with rich algebraic properties which
are constructed by means of the Hamiltonian Floer theory and which conve-
niently encode a part of the information contained in this theory. In general,
the definition of a partial symplectic quasi-state involves the choice of an
idempotent element in the commutative part QH•(M) of the quantum ho-
mology algebra of M . Though the default choice is just the unity of the
algebra, there exist some other meaningful choices, in particular in the case
when QH•(M) is semi-simple. This gives rise to another theme discussed in
this paper: “visible” topological obstructions to semi-simplicity (see Corol-
lary 1.24 and Theorem 1.25 below). For instance, we shall show that if a
monotone symplectic manifold M contains “too many” disjoint monotone
Lagrangian spheres whose minimal Maslov numbers exceed n+ 1, the quan-
tum homology QH•(M) cannot be semi-simple.
Let us pass to the precise set-up. For the reader’s convenience, the ma-
terial presented in this brief outline will be repeated in parts in the next
sections in a less compressed form.
1.2 Preliminaries on quantum homology
The Novikov Ring: Let F denote a base field which in our case will be
either C or Z2, and let Γ ⊂ R be a countable subgroup (with respect to the
addition). Let s, q be formal variables. Define a field KΓ whose elements are
generalized Laurent series in s of the following form:
KΓ :=
θ, zθ ∈ F , ♯
θ > c | zθ 6= 0
<∞, ∀c ∈ R
Define a ring ΛΓ := KΓ[q, q
−1] as the ring of polynomials in q, q−1 with
coefficients in KΓ. We turn ΛΓ into a graded ring by setting the degree of s
to be 0 and the degree of q to be 2.
The ring ΛΓ serves as an abstract model of the Novikov ring associated to
a symplectic manifold. Let (M,ω) be a closed connected symplectic manifold.
Denote by HS2 (M) the subgroup of spherical homology classes in the integral
homology group H2(M ;Z). Abusing the notation we will write ω(A), c1(A)
for the results of evaluation of the cohomology classes [ω] and c1(M) on
A ∈ H2(M ;Z). Set
π̄2(M) := H
2 (M)/ ∼,
where by definition
A ∼ B iff ω(A) = ω(B) and c1(A) = c1(B).
Denote by Γ(M,ω) := [ω](HS2 (M)) ⊂ R the subgroup of periods of the
symplectic form on M on spherical homology classes. By definition, the
Novikov ring of a symplectic manifold (M,ω) is ΛΓ(M,ω). In what follows,
when (M,ω) is fixed, we abbreviate and write Γ, K and Λ instead of Γ(M,ω),
KΓ(M,ω) and ΛΓ(M,ω) respectively.
Quantum homology: Set 2n = dimM . The quantum homology QH∗(M)
is defined as follows. First, it is a graded module over Λ given by
QH∗(M) := H∗(M ;F)⊗F Λ,
with the grading defined by the gradings on H∗(M ;F) and Λ:
deg (a⊗ zsθqk) := deg (a) + 2k .
Second, and most important, QH∗(M) is equipped with a quantum prod-
uct: if a ∈ Hk(M ;F), b ∈ Hl(M ;F), their quantum product is a class
a ∗ b ∈ QHk+l−2n(M), defined by
a ∗ b =
A∈π̄2(M)
(a ∗ b)A ⊗ s
−ω(A)q−c1(A),
where (a ∗ b)A ∈ Hk+l−2n+2c1(A)(M) is defined by the requirement
(a ∗ b)A ◦ c = GW
A (a, b, c) ∀c ∈ H∗(M ;F).
Here ◦ stands for the intersection index and GWFA (a, b, c) ∈ F denotes the
Gromov-Witten invariant which, roughly speaking, counts the number of
pseudo-holomorphic spheres inM in the class A that meet cycles representing
a, b, c ∈ H∗(M ;F) (see [55], [56], [41] for the precise definition).
Extending this definition by Λ-linearity to the whole QH∗(M) one gets
a correctly defined graded-commutative associative product operation ∗ on
QH∗(M) which is a deformation of the classical ∩-product in singular ho-
mology [37], [41], [55], [56], [69]. The quantum homology algebra QH∗(M)
is a ring whose unity is the fundamental class [M ] and which is a module
of finite rank over Λ. If a, b ∈ QH∗(M) have graded degrees deg (a), deg (b)
deg (a ∗ b) = deg (a) + deg (b)− 2n. (1)
We will be mostly interested in the commutative part of the quantum
homology ring (which in the case F = Z2 is, of course, the whole quantum
homology ring). For this purpose we introduce the following notation:
We denote by QH•(M) the whole quantum homology QH∗(M) if
F = Z2 and the even-degree part of QH∗(M) if F = C.
In general, given a topological space X, we denote by H•(X ;F) the
whole singular homology group H∗(X ;F) if F = Z2 and the even-
degree part of H∗(X ;F) if F = C.
Thus, in our notation the ring QH•(M) = H•(M ;F)⊗F Λ is always a com-
mutative subring with unity of QH∗(M) and a module of finite rank over Λ.
We will identify Λ with a subring of QH•(M) by λ 7→ [M ]⊗ λ.
1.3 An hierarchy of rigid subsets within Floer theory
Fix a non-zero idempotent a ∈ QH2n(M) (by obvious grading considera-
tions the degree of every idempotent equals 2n). We shall deal with spectral
invariants c(a,H), where H = Ht : M → R, t ∈ R, is a smooth time-
dependent and 1-periodic in time Hamiltonian function on M , or c(a, φH),
where φH is an element of the universal cover H̃am (M) of Ham(M) rep-
resented by an identity-based path given by the time-1 Hamiltonian flow
generated by H . If H is normalized, meaning that
dimM/2 = 0 for all
t, then c(a,H) = c(a, φH). These invariants, which nowadays are standard
objects of the Floer theory, were introduced in [45] (cf. [59] in the aspherical
case; also see [42],[43] for an earlier version of the construction and [22] for a
summary of definitions and results in the monotone case).
Disclaimer: Throughout the paper we tacitly assume that (M,ω) (as well
as (M ×T2, ω̄), when we speak of stable displaceability) belongs to the class
S of closed symplectic manifolds for which the spectral invariants are well
defined and enjoy the standard list of properties (see e.g. [41, Theorem
12.4.4]). For instance, S contains all symplectically aspherical and spherically
monotone manifolds. Furthermore, S contains all symplectic manifolds M2n
for which, on one hand, either c1 = 0 or the minimal Chern number (on
HS2 (M)) is at least n − 1 and, on the other hand, [ω](H
2 (M)) is a discrete
subgroup of R (cf. [64]). The general belief is that the class S includes all
symplectic manifolds.
Define a functional ζ : C∞(M) → R by
ζ(H) := lim
c(a, lH)
It is shown in [23] that the functional ζ has some very special algebraic
properties (see Theorem 3.6) which form the axioms of a partial symplectic
quasi-state introduced in [23]. The next definition is motivated in part by
the work of Albers [2].
Definition 1.1. A closed subset X ⊂ M is called heavy (with respect to ζ
or with respect to a used to define ζ) if
ζ(H) ≥ inf
H ∀H ∈ C∞(M) , (3)
and is called superheavy (with respect to ζ or a) if
ζ(H) ≤ sup
H ∀H ∈ C∞(M) . (4)
The default choice of an idempotent a is the unity [M ] ∈ QH∗(M). In this
case, as we shall see below, the collections of heavy and superheavy sets
satisfy the properties listed in Section 1.1 and include the examples therein.
In view of potential applications (including geometric obstructions to semi-
simplicity of the quantum homology), we shall work, whenever possible, with
general idempotents.
The asymmetry between supX H and infX H is related to the fact that
the spectral numbers satisfy a triangle inequality c(a ∗ b, φFφG) ≤ c(a, φF ) +
c(b, φG), while there may not be a suitable inequality “in the opposite direc-
tion”. In the case when such an “opposite” inequality exists (e.g. when a = b
is an idempotent and ζ defined by it is a genuine symplectic quasi-state – see
Section 1.6 below) the symmetry between supX H and infX H gets restored
and the classes of heavy and superheavy sets coincide.
Let us emphasize that the notion of (super)heaviness depends on the
choice of a coefficient ring for the Floer theory. In this paper the coefficients
for the Floer theory will be either Z2 or C depending on the situation. Unless
otherwise stated, our results on (super)heavy subsets are valid for any choice
the coefficients.
The group Symp (M) of all symplectomorphisms of M acts naturally on
H∗(M ;F) and hence on QH∗(M) = H∗(M ;F) ⊗F Λ. Clearly, the identity
component Symp0(M) of Symp (M) acts trivially on QH∗(M) and hence for
any idempotent a ∈ QH∗(M) the corresponding ζ is Symp0(M)-invariant.
Thus the image of a (super)heavy set under an element of Symp0(M) is again
a (super)heavy set with respect to the same idempotent a. If a is invariant
under the action of the whole Symp (M) (for instance, if a = [M ]) the classes
of heavy and superheavy sets with respect to a are invariant under the action
of the whole Symp (M) in agreement with the invariance property presented
in Section 1.1 above.
Let us mention also that the collections of (super)heavy sets enjoy a
stability property under inclusions: If X, Y , X ⊂ Y , are closed subsets of M
and X is heavy (respectively, superheavy) with respect to an idempotent a
then Y is also heavy (respectively, superheavy) with respect to the same a.
We are ready now to formulate the main results of the present section.
Theorem 1.2. Assume a and ζ are fixed. Then
(i) Every superheavy set is heavy, but, in general, not vice versa.
(ii) Every heavy subset is stably non-displaceable.
(iii) Every superheavy set intersects every heavy set. In particular, a super-
heavy set cannot be displaced by a symplectic (not necessarily Hamil-
tonian) isotopy and if the idempotent a is invariant under the symplec-
tomorphism group of (M,ω) (e.g. if a = [M ]), every superheavy set is
strongly non-displaceable.
The following theorem discusses the relation between heaviness/super-
heaviness properties with respect to different idempotents. In particular, it
shows that [M ] plays a special role among all the other non-zero idempotents
in QH∗(M).
Theorem 1.3. Assume a is a non-zero idempotent in the quantum homology.
(i) Every set that is superheavy with respect to [M ] is also superheavy with
respect to a.
(ii) Every set that is heavy with respect to a is also heavy with respect to
[M ].
(iii) Assume that the idempotent a is a sum of non-zero idempotents
e1, . . . , el and assume that a closed subset X ⊂ M is heavy with re-
spect to a. Then X is heavy with respect to ei for at least one i.
The next proposition shows that, in general, the heaviness of a set does
depend on the choice of an idempotent in the quantum homology.
Proposition 1.4. Consider the torus T2n equipped with the standard sym-
plectic structure ω = dp∧dq. Let M2n = T2n♯CP n be a symplectic blow-up of
T2n at one point (the blow up is performed in a small ball around the point).
Assume that the Lagrangian torus L ⊂ T2n given by q = 0 does not intersect
the ball in T2n, where the blow up was performed.
Then the proper transform of L (identified with L) is a Lagrangian sub-
manifold of M , which is not heavy with respect to some non-zero idempotent
a ∈ QH∗(M) but heavy with respect to [M ]. (Here we work with F = Z2).
Next, consider direct products of (super)heavy sets. We start with the fol-
lowing convention on tensor products. Let Γi, i = 1, 2, be two countable
subgroups of R. Let Ei be a module over KΓi. We put
E1⊗̂KE2 =
E1 ⊗KΓ1 KΓ1+Γ2
⊗KΓ1+Γ2
E2 ⊗KΓ2 KΓ1+Γ2
. (5)
If E1, E2 are also rings we automatically assume that the middle tensor prod-
uct is the tensor product of rings. In simple words, we extend both modules
to KΓ1+Γ2-modules and consider the usual tensor product over KΓ1+Γ2 .
Given two symplectic manifolds, (M1, ω1) and (M2, ω2), note that the
subgroups of periods of the symplectic forms satisfy
Γ(M1 ×M2, ω1 ⊕ ω2) = Γ(M1, ω1) + Γ(M2, ω2) .
Furthermore, due to the Künneth formula for quantum homology (see e.g.
[41, Exercise 11.1.15] for the statement in the monotone case; the general
case in our algebraic setup can be treated similarly) there exists a natural
ring monomorphism linear over KΓ1+Γ2
QH2n1(M1)⊗̂KQH2n2(M2) →֒ QH2n1+2n2(M1 ×M2) ,
We shall fix a pair of idempotents ai ∈ QH∗(Mi), i = 1, 2. The notions
of (super)heaviness in M1,M2 and M1 ×M2 are understood in the sense of
idempotents a1, a2 and a1 ⊗ a2 respectively.
Theorem 1.5. Assume that Xi is a heavy (resp. superheavy) subset of Mi
with respect to some idempotent ai, i = 1, 2. Then the product X1 × X2
is a heavy (resp. superheavy) subset of M with respect to the idempotent
a1 ⊗ a2 ∈ QH•(M1 ×M2).
An important class of superheavy sets is given by stable stems introduced
and illustrated in Section 1.1.
Theorem 1.6. Every stable stem is a superheavy subset with respect to any
non-zero idempotent a ∈ QH∗(M). In particular, it is strongly and stably
non-displaceable.
In the next section we present an example of stable stems coming from Hamil-
tonian torus actions.
1.4 Hamiltonian torus actions
Fibers of the moment maps of Hamiltonian torus actions form an interesting
playground for testing the various notions of displaceability and heaviness
introduced above. Throughout the paper we deal with effective actions only,
that is we assume that the map ϕ : Tk → Ham(M) defining the action
is a monomorphism. Furthermore, we assume that the moment map Φ =
(Φ1, . . . ,Φk) : M → R
k of the action is normalized: Φi is a normalized
Hamiltonian for all i = 1, . . . , k. By the Atiyah-Guillemin-Sternberg theorem
[6], [30], the image ∆ = Φ(M) of Φ is a k-dimensional convex polytope,
called the moment polytope. The subsets Φ−1(p), p ∈ ∆, are called fibers of
the moment map. A torus action is called compressible if the image of the
homomorphism ϕ♯ : π1(T
k) → π1(Ham(M)), induced by the action ϕ, is a
finite group.
Theorem 1.7. Assume that (M,ω) is equipped with a compressible Hamilto-
nian Tk-action with moment map Φ and moment polytope ∆. Let Y ⊂ ∆ be
any closed convex subset which does not contain 0. Then the subset Φ−1(Y )
is stably displaceable. In particular, the fiber Φ−1(0) is a stable stem.
Note that for symplectic toric manifolds, that is when 2k = dimM , the point
0 is the barycenter of the moment polytope with respect to the Lebesgue
measure. This follows from our assumption on the normalization of the
moment map.
Theorems 1.6 and 1.7 imply that the fiber Φ−1(0) of a compressible torus
action is stably non-displaceable, and thus we get the complete description
of stably displaceable fibers for such actions.
In the case when the action is not compressible, the question of the com-
plete description of stably non-displaceable fibers remains open. We make a
partial progress in this direction by presenting at least one such fiber, called
the special fiber, explicitly in the case when (M,ω) is spherically monotone:
[ω]|HS2 (M)
= κ c1(TM)|HS2 (M)
, κ > 0 .
The special fiber can be described via the mixed action-Maslov homomor-
phism introduced in [49]: Let (M2n, ω) be a spherically monotone symplectic
manifold, and let {ft}, t ∈ [0, 1], be any loop of Hamiltonian diffeomorphisms,
with f0 = f1 = 1, generated by a 1-periodic normalized Hamiltonian func-
tion F (x, t). The orbits of any Hamiltonian loop are contractible due to the
standard Floer theory1. Pick any point x ∈ M and any disc u : D2 → M
spanning the orbit γ = {ftx}. Define the action
2 of the orbit by
AF (γ, u) :=
F (γ(t), t)dt−
u∗ω .
Trivialize the symplectic vector bundle u∗(TM) over D2 and denote by
mF (γ, u) the Maslov index of the loop of symplectic matrices corresponding
to {ft∗} with respect to the chosen trivialization. One readily checks that,
in view of the spherical monotonicity, the quantity
I(F ) := −AF (γ, u)−
mF (γ, u)
does not depend on the choice of the point x and the disc u, and is invariant
under homotopies of the Hamiltonian loop {ft}. In fact, I is a well defined
homomorphism from π1(Ham(M)) to R (see [49], [68]).
Assume again that ϕ : Tk → Ham(M,ω) is a Hamiltonian torus ac-
tion. Write ϕ♯ for the induced homomorphism of the fundamental groups.
Since the target space Rk of the moment map Φ is naturally identified with
Hom(π1(T
k),R), the pull back −ϕ∗♯I of the mixed action-Maslov homomor-
phism with the reversed sign can be considered as a point of Rk. We call
it a special point and denote by pspec. The preimage Φ
−1(pspec) is called the
special fiber of the moment map. In the case k = 1, when Φ is a real-valued
function on M , we will call pspec the special value of Φ.
1The Floer theory guarantees the existence of at least one contractible periodic orbit –
this is not obvious a priori if {ft} is not an autonomous flow. Since all the orbits of {ft}
are homotopic, all of them are contractible.
2Note that our action functional and the one in [49] are of opposite signs.
If k = n and M is a symplectic toric manifold, then pspec can be defined
in purely combinatorial terms involving only the polytope ∆. Namely, pick
a vertex x of ∆. Since ∆ in this case is a Delzant polytope [20], there is a
unique (up to a permutation) choice of vectors v1, . . . ,vn which
• originate at x;
• span the n rays containing the edges of ∆ adjacent to x;
• form a basis of Zn over Z.
Proposition 1.8.
pspec = x+ κ
vi. (6)
Proof. The vertices of the moment polytope are in one-to-one correspondence
with the fixed points of the action. Let x ∈ M be the fixed point corre-
sponding to the vertex x = (x1, . . . ,xn). Then the vectors vj = (v
j , . . . , v
j = 1, . . . , n, are simply the weights of the isotropy Tn-action on TxM . Since
the definition of the mixed action-Maslov invariant of a Hamiltonian circle
action does not depend on the choice of a 1-periodic orbit and a disc span-
ning it, let us compute all Ii, l = 1, . . . , n, using the constant periodic orbit
concentrated at the fixed point x and the constant disc u spanning it. Clearly,
AΦi(x, u) = Φi(x) = xi and mΦi(x, u) = 2
vij ∀i = 1, . . . , n,
which readily yields formula (6).
E.Shelukhin pointed out to us that by summing up equations (6) over all the
vertices x(1), . . . ,x(m) ∈ Rn of the moment polytope, one readily gets that
pspec =
Theorem 1.9. Assume M2n is a spherically monotone symplectic manifold
equipped with a Hamiltonian Tk-action. Then the special fiber of the moment
map is superheavy with respect to any (non-zero) idempotent a ∈ QH2n(M).
In particular, it is stably and strongly non-displaceable.
Let us mention that, in particular, the special fiber is non-empty and so
pspec ∈ ∆. Moreover pspec is an interior point of ∆ – otherwise Φ
−1(pspec) is
isotropic of dimension < n and hence displaceable (see e.g. [9]).
Remark 1.10. If dimM = 2dimTk (that is we deal with a symplectic toric
manifold), the special fiber, say L, is a Lagrangian torus. In fact, this torus
is monotone: for every D ∈ π2(M,L) we have
ω = κ ·mL(D) ,
where mL stands for the Maslov class of L. This is an immediate consequence
of the definitions.
Remark 1.11. Note that when M is spherically monotone and the action is
compressible Theorems 1.7 and 1.9 match each other: in this case pspec = 0
and therefore the special fiber is a stable stem by Theorem 1.7. It is unknown
whether this property persists for the special fibers of non-compressible ac-
tions.
Example 1.12. Let M be the monotone symplectic blow up of CP 2 at k
points (0 ≤ k ≤ 3) which is equivariant with respect to the standard T2-
action and which is performed away from the Clifford torus in CP 2. Since
the blow-up is equivariant, M comes equipped with a Hamiltonian T2-action
extending the T2-action on CP 2. The Clifford torus is a fiber of the moment
map of the T2-action on CP 2. Let L ⊂M be the Lagrangian torus which is
the proper transform of the Clifford torus under the blow-up – it is a fiber of
the moment map of the T2-action on M . Using Proposition 1.8 it is easy to
see that L is the special fiber of M . According to Theorem 1.9, it is stably
and strongly non-displaceable. In fact, it is a stem: the displaceability of
all the other fibers was checked for k = 0 in [10], for k = 1 in [23] and for
k = 2, 3 in [40].
We refer to Section 1.7.2 for further discussion of related problems and very
recent advances.
Digression: Calabi vs. action-Maslov. The method used to prove
Theorem 1.9 also allows to prove the following result involving the mixed
action-Maslov homomorphism. Denote by vol (M) the symplectic volume of
M . Consider the function µ : H̃am (M) → R defined by
µ(φH) := −vol (M) lim
c(a, φlH)/l.
In the case when a is the unity in a field that is a direct summand in the
decomposition of the K-algebra QH2n(M,ω), as an algebra, into a direct
sum of subalgebras, µ is a homogeneous quasi-morphism on H̃am (M) called
Calabi quasi-morphism [22],[24],[46]; in the general case it has weaker prop-
erties [23]. With this language the functional ζ (on normalized functions) is
induced (up to a constant factor) by the pull-back of µ to the Lie algebra of
H̃am (M).
Following P.Seidel we described in [22] the restriction of µ (in fact, for any
spherically monotoneM) on π1(Ham(M)) ⊂ H̃am (M) in terms of the Seidel
homomorphism π1(Ham(M)) → QH
∗ (M), where QH
∗ (M) denotes the
group of invertible elements in the ring QH∗(M). Here we give an alternative
description of µ|π1(Ham(M)) in terms of the mixed action-Maslov homomor-
phism I which, in turn, also provides certain information about the Seidel
homomorphism.
Theorem 1.13. Assume M is spherically monotone and let µ be defined as
above for some non-zero idempotent a ∈ QH∗(M). Then
µ|π1(Ham(M)) = vol (M) · I.
Note that, in particular, µ|π1(Ham(M)) does not depend on a used to de-
fine µ. The theorem also implies that µ descends to a quasi-morphism on
Ham(M) if and only if I : π1(Ham(M)) → R vanishes identically (since µ
descends to a quasi-morphism on Ham(M) if and only if µ|π1(Ham(M)) ≡ 0 –
see e.g. [22], Prop. 3.4). The proof of the theorem is given in Section 9.1.
Let us mention also that, interestingly enough, the homomorphism I
coincides with the restriction to π1(Ham(M)) of yet another quasi-morphism
on H̃am (M) constructed by P.Py (see [52, 53]).
Digression: Action-Maslov homomorphism and Futaki invari-
ant. This remark grew from an observation pointed out to us by Chris
Woodward – we are grateful to him for that. Assume that our symplectic
manifoldM is complex Kähler (i.e. the symplectic structure onM is induced
by the Kähler one) and Fano (by this we mean here that [ω] = c1). Assume
also that a Hamiltonian S1-action {ft} preserves the Kähler metric and the
complex structure. For instance, if M2n is a symplectic toric manifold it can
be equipped canonically with a complex structure and a Kähler metric invari-
ant under the Tn-action on M , hence under the action of any S1-subgroup
{ft} of T
Let V be the Hamiltonian vector field generating the Hamiltonian flow
{ft}. Since {ft} preserves the complex structure, one can associate to V its
Futaki invariant F(V ) ∈ C [29]. It has been checked by E.Shelukhin [63]
that, up to a universal constant factor, this Futaki invariant is equal to the
value of the mixed action-Maslov homomorphism on the loop {ft}:
F(V ) = const · I({ft}).
Note that if such an M admits a Kähler-Einstein metric then the Futaki
invariant has to vanish [29] – thus if I({ft}) 6= 0 the manifold does not admit
a Kähler-Einstein metric. Moreover, if M2n is toric the opposite is also true:
if the Futaki invariant vanishes for any V generating a subgroup of the torus
Tn acting onM thenM admits a Kähler-Einstein metric – this follows from a
theorem by Wang and Zhu [67], combined with a previous result of Mabuchi
[38]. In terms of the moment polytope, the vanishing of the Futaki invariant,
and accordingly the existence of a Kähler-Einstein metric, on a Kähler Fano
toric manifold means precisely that the special point of the polytope coincides
with the barycenter.
1.5 Super(heavy) monotone Lagrangian submanifolds
Let (M2n, ω) be a closed spherically monotone symplectic manifold with [ω] =
κ · c1(TM) on π2(M), κ > 0. Let L ⊂ M be a closed monotone Lagrangian
submanifold with the minimal Maslov number NL ≥ 2. As usually, we put
NL = +∞ if π2(M,L) = 0. As before, we work with the basic field F which
is either Z2 or C. In the case F = C, we assume that L is relatively spin, that
is L is orientable and the 2nd Stiefel-Whitney class of L is the restriction of
some integral cohomology class of M .
Disclaimer: In the case F = C the results of this section are conditional:
We take for granted that Proposition 8.1 below, which was proved by Biran
and Cornea [15] for homologies with Z2-coefficients, extends to homologies
with C-coefficients. In each of the specific examples below we will explicitly
state which F we are using and whenever we use F = C we assume that L
is relatively spin.
Denote by j the natural morphism j : H•(L;F) → H•(M ;F). We say that
L satisfies the Albers condition [2] if there exists an element S ∈ H•(L;F) so
that j(S) 6= 0 and
deg S > dimL+ 1−NL .
We shall refer to such S as to an Albers element of L.
Example 1.14. Assume [L] ∈ H•(L;F) and j([L]) ∈ H•(M ;F) is non-zero.
This means precisely that [L] is an Albers element of L.
A closed monotone Lagrangian submanifold L which satisfies this con-
dition (and whose minimal Maslov number is greater than 1) will be called
homologically non-trivial in M .
Theorem 1.15. Let L be a closed monotone Lagrangian submanifold satisfy-
ing the Albers condition. Then L is heavy with respect to [M ]. In particular,
any homologically non-trivial Lagrangian submanifold is heavy with respect
to [M ].
Example 1.16. Assume that π2(M,L) = 0. Then the homology class of a
point is an Albers element of L, and hence L is heavy. Note that in this
case heaviness cannot be improved to superheaviness: the meridian on the
two-torus is heavy but not superheavy. Here we took F = Z2.
Example 1.17 (Lagrangian spheres in Fermat hypersurfaces). More exam-
ples of heavy (but not necessarily superheavy) monotone Lagrangian sub-
manifolds can be constructed as follows3.
Let M ⊂ CP n+1 be a smooth complex hypersurface of degree d. The
pull-back of the standard symplectic structure from CP n+1 turns M into a
symplectic manifold (of real dimension 2n). If d ≥ 2, then, as it is explained,
for instance, in [12],M contains a Lagrangian sphere: M can be included into
a family of algebraic hypersurfaces of CP n+1 with quadratic degenerations at
isolated points and the vanishing cycle of such a degeneration can be realized
by a Lagrangian sphere following [5], [21], [60], [61], [62].
Let M ⊂ CP n+1 be a projective hypersurface of degree d, 2 ≤ d < n+ 2.
The minimal Chern number of M equals N := n+2− d > 0. Let Ln ⊂ M2n
be a simply connected Lagrangian submanifold (for instance, a Lagrangian
sphere).
First, consider the case when n is even, L is relatively spin and the Euler
characteristics of L does not vanish (this is the case for a sphere). Then the
3We thank P.Biran for his indispensable help with these examples.
homology class j([L]) ∈ Hn(M ;Z) is non-zero: its self-intersection number
in M up to the sign equals the Euler characteristic. Thus [L] is an Albers
element. (Here we use F = C). In view of Theorem 1.15, L is heavy with
respect to [M ].
Second, suppose that n is of arbitrary parity but n > 2d − 3, and no
restriction on the Euler characteristics of L is assumed anymore. This yields
NL = 2N > n+ 1 and thus L satisfies the Albers condition with the class of
a point P as an Albers element. Thus L is heavy with respect to [M ] – here
we use F = Z2.
Finally, fix n ≥ 3 and an even number d such that 4 ≤ d < n+2. Consider
a Fermat hypersurface of degree d
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
n+1 .
Its real part L := M ∩ RP n+1 lies in the affine chart z0 6= 0 and is given by
the equation
xd1 + . . .+ x
n+1 = 1,
where xj := Re(zj/z0) . Since d is even, L is an n-dimensional sphere. As
it was explained above, L is heavy with respect to [M ] if either n is even
(and F = C) or n > 2d − 3 (and F = Z2). However, in either case L is not
superheavy with respect to [M ]. Indeed, let Σd ≈ Zd be the group of complex
roots of unity. Given a vector α = (α1, . . . , αn) ∈ (Σd)
n+1, denote by fα the
symplectomorphism of M given by
fα(z0 : z1 : . . . : zn+1) = (z0 : α1z1 : . . . : αn+1zn+1) . (7)
If all αj ∈ C\R, then αjx /∈ R whenever x ∈ R\{0}, and thus fα(L)∩L = ∅.
Therefore L is strongly displaceable and the claim follows from the part (iii)
of Theorem 1.2.
The next result gives a user-friendly sufficient condition of superheaviness.
Theorem 1.18. Assume L is homologically non-trivial in M and assume
a ∈ QH2n(M) is a non-zero idempotent divisible by j([L]) in QH•(M), that
is a ∈ j([L]) ∗QH•(M). Then L is superheavy with respect to a.
The homological non-triviality of L in the hypothesis of the theorem means
just that [L] is an Albers element of L (see Example 1.14). In fact, the
theorem can be generalized to the cases when L has other Albers elements –
see Remark 8.3 (ii).
Example 1.19 (Lagrangian spheres in quadrics). Here we work with F = C.
Let M be the real part of the Fermat quadric M = {−z20 +
j=1 z
j = 0}.
Assume that n is even and L is a simply connected Lagrangian submanifold
with non-vanishing Euler characteristic (e.g. a Lagrangian sphere). Under
this assumption, [L] ∈ H•(L) and j([L]) 6= 0, since L has non-vanishing self-
intersection. Denote by p ∈ H∗(M ;F) the class of a point. The quantum
homology ring of M was described by Beauville in [8]. In particular, p ∗ p =
w−2[M ], where w = sκnqn. Thus
a± :=
[M ]± pw
are idempotents. One can show that j([L]) divides a− and hence L is a−-
superheavy. Since a− is invariant under the action of Symp(M), the manifold
L is strongly non-displaceable.
For simplicity, we present the calculation in the case n = 2 – the general
case is absolutely analogous. The 2-dimensional quadric is symplectomorphic
to (S2 × S2, ω ⊕ ω). Denote by A and B the classes of [S2] × [point] and
[point] × [S2] respectively. Since the symplectic form vanishes on j([L]) we
get that j([L]) = l(B − A) with l 6= 0. It is known that A ∗ B = p and
B ∗B = w−1[M ]. Thus j([L]) ∗ 1
wB = a−, that is j([L]) divides a−.
In particular, the Lagrangian anti-diagonal
∆ := {(x, y) ∈ S2 × S2 : x = −y} ,
which is diffeomorphic to the 2-sphere, is superheavy with respect to a−. It is
unknown whether ∆ is super-heavy with respect to a+. Further information
on superheavy Lagrangian submanifolds in the quadrics can be extracted
from [15].
Example 1.20 (A non-heavy monotone Lagrangian torus in S2 × S2). Con-
sider the quadric M = S2 × S2 from the previous example. We will think of
S2 as of the unit sphere in R3 whose symplectic form is the area form divided
by 4π. We will work again with F = C. Interestingly enough, such an M
contains a monotone Lagrangian torus that is not heavy with respect to a−.
Namely, consider a submanifold K given by equations4
K = {(x, y) ∈ S2 × S2 : x1y1 + x2y2 + x3y3 = −
, x3 + y3 = 0} .
4We thank Frol Zapolsky for his help with calculations in this example.
One readily checks that K is a monotone Lagrangian torus with NK = 2
which represents a zero element inH2(M ;F) (both with F = C and F = Z2).
Thus H•(K;F) does not contain any Albers element. Furthermore, K is
disjoint from the Lagrangian anti-diagonal ∆ and hence is not heavy with
respect to a− since, as it was shown above, ∆ is superheavy with respect to
a−. In particular, K is an exotic monotone torus: it is not symplectomorphic
to the Clifford torus which is a stem and hence a−-superheavy. A further
study of exotic tori in products of spheres is currently being carried out by
Y.Chekanov and F.Schlenk.
It is an interesting problem to understand whether K is superheavy with
respect to a+, or at least non-displaceable. Identify M \ {the diagonal} with
the unit co-ball bundle of the 2-sphere. After such an identification ∆ corre-
sponds to the zero section, while K corresponds to a monotone Lagrangian
torus, say K ′. Interestingly enough, the Lagrangian Floer homology of K ′
in T ∗S2 (with F = Z2) does not vanish as was shown by Albers and Frauen-
felder in [3], and thus K is not displaceable in M \ {the diagonal}. Thus
the question on (non)-displaceability of K is related to understanding of the
effect of the compactification of the unit co-ball bundle to S2 × S2.
The proofs of theorems above are based on spectral estimates due to
Albers [2] and Biran-Cornea [15]. Furthermore, the results above admit
various generalizations in the framework of Biran-Cornea theory of quantum
invariants for monotone Lagrangian submanifolds, see [15] and the discussion
in Section 8 below.
1.6 An effect of semi-simplicity
Recall that a commutative (finite-dimensional) algebra Q over a field A is
called semi-simple if it splits into a direct sum of fields as follows: Q =
Q1 ⊕ . . .⊕Qd , where
• each Qi ⊂ Q is a finite-dimensional linear subspace over A;
• each Qi is a field with respect to the induced ring structure;
• the multiplication in Q respects the splitting:
(a1, . . . , ad) · (b1, . . . , bd) = (a1b1, . . . , adbd).
A classical theorem of Wedderburn (see e.g. [66], §96) implies that the semi-
simplicity is equivalent to the absence of nilpotents in the algebra.
Remark 1.21. Assume that the K-algebra QH2n(M,ω) splits, as an algebra,
into a direct sum of two algebras, at least one of which is a field, and let e
be the unity in that field. In particular, this is the case when QH2n(M,ω) =
Q1⊕ . . .⊕Qd is semi-simple and e is the unity in one of the fields Qi. A slight
generalization of the argument in [23, 46] (see [24], the remark on pp. 56-57)
shows that the partial quasi-state ζ(e, ·) associated to e is R-homogeneous
(and not just R+-homogeneous as in the general case).
This immediately yields that every set which is heavy with respect to e is
automatically superheavy with respect to e.
In fact, in this situation ζ is a genuine symplectic quasi-state in the sense of
[23] and, in particular, a topological quasi-state in the sense of Aarnes [1] (see
[23] for details). In [1] Aarnes proved an analogue of the Riesz representation
theorem for topological quasi-states which generalizes the correspondence
between genuine states (that is positive linear functionals on C(M)) and
measures. The object τζ corresponding to a quasi-state ζ is called a quasi-
measure (or a topological measure). With this language in place, the sets that
are (super)heavy with respect to ζ are nothing else but the closed sets of the
full quasi-measure τζ . Any two such sets have to intersect for the following
basic reason: any quasi-measure is finitely additive on disjoint closed subsets
and therefore if two closed subsets of M of the full quasi-measure do not
intersect, the quasi-measure of their union must be greater than the total
quasi-measure of M , which is impossible.
Example 1.22. In this example we again assume that F = Z2. Let M =
CP n be equipped with the Fubini-Study symplectic structure ω, normalized
so that [ω] = c1, and let A ∈ H2n−2(M) be the homology class of the hyper-
plane. One readily verifies the following K-algebra isomorphism
QH2n(M) ∼= K[X ]/〈X
n+1 − u−1〉,
where
K = Z2[[u] = {zku
k + zk−1u
k−1 + . . . , zi ∈ Z2 ∀i}
is the field of Laurent-type series in u := sn+1 with coefficients in Z2 and
X = qA. Since no root of degree 2 or more of u−1 is contained in K, the
polynomial P is irreducible over K for any n (see e.g. [34], Theorem 9.1) and
therefore QH2n(M) is a field. Hence the collections of heavy and superheavy
sets with respect to the fundamental class coincide.
We claim that L := RP n ⊂ CP n is superheavy. The case n = 1 cor-
responds to the equator of the sphere, which is known to be a stable stem.
For n ≥ 2, note that NL = n + 1 and S = [RP
2] is an Albers element of L.
Therefore, L is [M ]-heavy by Theorem 1.15, and hence superheavy.
The next result follows directly from Theorem 1.3 (iii) and Remark 1.21:
Theorem 1.23. Assume that QH2n(M) is semi-simple and splits into a
direct sum of d fields whose unities will be denoted by e1, . . . , ed. Assume that
a closed subset X ⊂M is heavy with respect to a non-zero idempotent a – as
one can easily see, such an idempotent has to be of the form a = ej1+ . . .+ejl
for some 1 ≤ j1 < . . . < jl ≤ d. Then X is superheavy with respect to some
eji, 1 ≤ i ≤ l.
The theorem yields the following geometric characterization of non-semi-
simplicity of QH2n(M). Namely, define the symplectic Torelli group as the
group of all symplectomorphisms of M which induce the identity map on
H•(M ;F). For instance, this group contains Symp0(M). Note that any ele-
ment of the symplectic Torelli group acts trivially on the quantum homology
of M and hence maps sets (super)heavy with respect to an idempotent a to
sets (super)heavy with respect to a.
Now Theorem 1.23 readily implies the following
Corollary 1.24. Assume that (M,ω) contains a closed subset X which is
heavy with respect to a non-zero idempotent and displaceable by a symplec-
tomorphism from the symplectic Torelli group. Then QH2n(M) is not semi-
simple.
The simplest examples are provided by sets of the form X×{a meridian}
in M × T2 with a heavy X .
Another result in the same vein is as follows5. Given a set Y of positive
integers, put βY (M) =
i∈Y βi(M), where βi(M) stands for the i-th Betti
number of M over F .
5In the case F = C, Theorem 1.25 is conditional, see the disclaimer in the previous
section.
Theorem 1.25. Assume that either of the following (not mutually excluding)
conditions holds:
(a) M contains m > βY (M) + 1 pair-wise disjoint closed monotone La-
grangian submanifolds whose minimal Maslov numbers are greater than n+1
and belong to a set Y of positive integers.
(b) M contains pair-wise disjoint homologically non-trivial Lagrangian sub-
manifolds6 whose fundamental classes, viewed as (non-zero) elements of
H•(M ;F), are linearly dependent over F .
(In the case F = C assume that all the Lagrangian submanifolds above are
also relatively spin.)
Then QH2n(M) is not semi-simple.
The proof is given in Section 8.
Example 1.26. For instance, if all the Lagrangian submanifolds from part
(a) of the theorem are simply connected, their minimal Maslov numbers are
equal to 2N , so that the set Y consists of one element: Y = {2N}. Thus
if 2N > n + 1 and QH2n(M) is semi-simple, M cannot contain more than
β2N(M)+ 1 pair-wise disjoint simply-connected Lagrangians (provided all of
them are relatively spin if we work with F = C).
Example 1.27. Set F = C. Fix n ≥ 11 and an even number d such that
6 ≤ d < (n + 3)/2. Consider a Fermat hypersurface of degree d
M = {−zd0 + z
1 + . . .+ z
n+1 = 0} ⊂ CP
As we already saw in Example 1.17, the manifold L := M ∩ RP n+1 is an
n-dimensional Lagrangian sphere. Consider the images fα(L), where sym-
plectomorphisms fα are defined by (7). Note that, as long as αj/βj 6= ±1
for all j, the Lagrangian spheres fα(L) and fβ(L) are disjoint. Using this
observation, it is easy to find d/2 disjoint Lagrangian spheres in M .
The minimal Chern number N ofM equals n+2−d, and so 2N lies in the
interval [n+2, 2n−4]. In this case β2N(M) = 1 (see e.g. [31]). Since d/2 > 2,
we conclude from the previous example that QH2n(M) is not semi-simple.
This conclusion agrees with the computation of QH∗(M) by Beauville [8].
6See Example 1.14 for the definition. As in that example we again assume that all our
Lagrangian submanifolds are closed, monotone and have minimal Maslov number greater
than 1.
It would be interesting to find examples of symplectic manifolds where the
quantum homology is not known a priori and where the above theorems are
applicable. Let us mention that different obstructions to the semi-simplicity
of QH•(M) coming from Lagrangian submanifolds were recently found by
Biran and Cornea [14].
1.7 Discussion and open questions
1.7.1 Strong displaceability beyond Floer theory?
Clearly, displaceability implies stable displaceability. The converse is not
true, as the next example shows:
Example 1.28. Consider the complex projective space CP n equipped with
the Fubini-Study symplectic form (in our normalization the area of a line
equals 1). Identify CP n with the symplectic cut of the Euclidean ball B(1) ⊂
Cn (that is the boundary of B(1) is collapsed to CP n−1 along the fibers of
the Hopf fibration, see [36]), where B(r) := {π|z|2 ≤ r}. Then B(r) ⊂ CP n
(i) displaceable for r < 1/2;
(ii) strongly non-displaceable but stably displaceable for r ∈ [1/2, n/n+1);
(iii) strongly and stably non-displaceable for r ≥ n/n+ 1.
It is instructive to analyze the techniques involved in the proofs: The strong
non-displaceability result in (ii) is an immediate consequence of Gromov’s
packing-by-two-balls theorem, which is proved via the J-holomorphic variant
of the theorem which states that there exists a J-holomorphic line in CP n
passing through any two points. In the case (iii) the ball B(r) contains the
Clifford torus, which is stably non-displaceable. This follows either from the
fact that the Clifford torus is a stem (see [10]), or from non-vanishing of its
Lagrangian Floer homology [16].
The displaceability of B(r) in (i) follows from the explicit construction
of the two balls packing (see [33]). The stable displaceability in (ii) is a
direct consequence of Theorem 1.7 above: Indeed, consider the standard Tn-
action on CP n. The normalized moment polytope ∆ ⊂ Rn has the form
∆ = ∆stand + w where ∆stand is the standard simplex {ρi ≥ 0,
ρi ≤ 1} in
Rn, where (ρ1, . . . , ρn) denote coordinates in R
n, and w = − 1
(1, . . . , 1).
Note that the ball B(r) equals to Φ−1(∆r) where ∆r := r ·∆stand + w. Note
that ∆r does not contain the origin exactly when r ≤
which yields the
stable displaceability in (ii) above.
A mysterious feature of Example 1.28 is as follows. On the one hand, we
believe in the following general empiric principle: whenever one can establish
the non-displaceability of a subset by means of the Floer homology theory,
one gets for free the stable non-displaceability. On the other hand, we be-
lieve, following a philosophical explanation provided by Biran, that Gromov’s
packing-by-two-balls theorem may be extracted from some “operations” in
Floer homology. Example 1.28 shows that at least one of these beliefs is
wrong. It would be interesting to clarify this issue.
1.7.2 Heavy fibers of Poisson-commutative subspaces
It was shown in [23] that for any finite-dimensional Poisson-commutative
subspace A ⊂ C∞(M) at least one of the fibers of its moment map Φ has to
be non-displaceable.
Question. Is it true that at least one fiber of Φ has to be heavy (with respect
to some non-zero idempotent a ∈ QH∗(M))?
It is easy to construct an example of A whose moment map Φ has no
superheavy fibers: take T2 with the coordinates p, q mod 1 on it and take A
to be the set of all smooth functions depending only on p – the corresponding
Φ defines the fibration of T2 by meridians none of which is superheavy.
Here is another question which concerns fibers of symplectic toric man-
ifolds, i.e. fibers of a moment map Φ of an effective Hamiltonian Tn-action
on (M2n, ω). Assume M is (spherically) monotone. Theorem 1.9 shows that
in such a case the special fiber ofM is superheavy, hence stably and strongly
non-displaceable. In all the examples where it has been checked this turns
out to be the only non-displaceable fiber of M .
Question. Is the special fiber for a monotone symplectic toric M always
a stem? In particular, is it the only non-displaceable fiber of the moment
In the monotone case the special fiber is clearly the only heavy fiber of
the moment map, because it is superheavy and any other heavy fiber would
have had to intersect it. On the other hand, if we consider a Hamiltonian Tk-
action on M2n with k < n there can be more than one non-displaceable fiber
of the moment map – for instance, because of purely topological obstructions:
the simplest Hamiltonian T1-action on CP 2 provides such an example. In
the case of monotone symplectic toric manifolds of dimension bigger than 4
the question above is absolutely open.
After the first draft of this paper appeared, a remarkable progress in this
direction has been achieved in the works by Cho [17] and Fukaya, Oh, Ohta
and Ono [28]: In particular, it turns out that a non-monotone symplectic
toric manifold can have more than one non-displaceable fiber – this happens
already for certain equivariant blowups of CP 2.
Organization of the paper:
In Section 2 we prove Theorem 1.7 which in particular states that the
special fiber of a compressible torus action is a stable stem.
In Section 3 we sum up various preliminaries from Floer theory including
basic properties of spectral invariants and partial symplectic quasi-states. In
addition we spell out a useful property of the Conley-Zehnder index: it is a
quasi-morphism on the universal cover of the symplectic group (see Propo-
sition 3.5). For completeness we extract a proof of this property from [54];
alternatively, one can use the results of [19].
In Section 4 we prove parts (i) and (iii) of Theorem 1.2 and Theorem 1.3
on basic properties of (super)heavy sets.
In Section 5 we prove Theorem 1.5 on products of (super)heavy sets. Our
approach is based on a quite general product formula for spectral invariants
(Theorem 5.1), which is proved by a fairly lengthy algebraic argument.
In Section 6 we prove Theorem 1.2 (ii) on stable non-displaceability of
heavy subsets. The argument involves a “baby version” of the above-men-
tioned product formula.
In Section 7 we prove superheaviness of stable stems.
In Section 8 we bring together the proofs of various results related to
(super)heaviness of monotone Lagrangian submanifolds satisfying the Albers
condition, including Theorems 1.15, 1.18, 1.25 and Proposition 1.4.
In Section 9 we prove Theorem 1.9 on superheaviness of special fibers of
Hamiltonian torus actions on monotone symplectic manifolds. The proof is
quite involved. In fact, two tricks enabled us to shorten our original argu-
ment: First, we use the Fourier transform on the space of rapidly decaying
functions on the Lie coalgebra of the torus in order to reduce the problem
to the case of Hamiltonian circle actions. Second, we systematically use the
quasi-morphism property of the Conley-Zehnder index for asymptotic calcu-
lations with Hamiltonian spectral invariants. Finally, in Section 9.1 we prove
Theorem 1.13.
Figure 1 sums up the hierarchy of the non-displaceability properties dis-
cussed above.
������������������������
MHierarchy of non−displaceability properties of a closed subset of 
Heavy
aidempotent 
Superheavy
idempotent  a
wrt a non−zero wrt a non−zero
(3) (4) (5) (6)
action on a spherically
torus action on a (not
a compressible Hamiltonian
necessarily monotone)
Special fiber of
monotone M
Always true.
True under certain conditions (see below)
? Question (under certain conditions − see below)
Monotone
Lagrangian
submanifold L
(14) (15)
(17) (18)
(21) (22)
(16b)
Product of codimension−1
skeletons of fine
triangulations
Strongly
non−displaceable
a Hamiltonian torus
Non−displaceable
a symplectic isotopy
Non−displaceable by
wrt [M]
Heavy
(16a)
Superheavy
wrt [M]
Stable stem
Stably
non−displaceable
Zero fiber of
Figure 1: Hierarchy of non-displaceability properties
(1),(2),(6),(19) - Trivial.
(3) True if a is invariant under the action of the whole group Symp (M) –
Theorem 1.2, part (iii).
(4), (9) Theorem 1.2, part (iii).
(5) True if the algebra QH2n(M) is semi-simple – see Corollary 1.24.
(7a) True if the algebra QH2n(M) splits, as an algebra, into a direct sum of
two algebras, at least one of which is a field, and a is the unity element in
that field – see Remark 1.21.
(7b), (16b) Theorem 1.2, part (i).
(8) Theorem 1.2, part (ii).
(10) Theorem 1.18 (see the assumptions on L there).
(11) True if the algebra QH2n(M) is semi-simple – see Corollary 1.24.
(12) Theorem 1.3, part (i).
(13) Theorem 1.3, part (ii).
(14) Theorem 1.18 (see the assumptions on L there) with a = [M ] – i.e. j(L)
is invertible in QH•(M).
(15) L satisfies the Albers condition – see Theorem 1.15.
(16a) True if QH2n(M) is a field – see Remark 1.21.
(17) Theorem 1.6.
(18) Theorem 1.9.
(20) Theorem 1.7.
(21) Is the special fiber for a monotone symplectic toric M always a stem?
See Section 1.7.2.
(22) True if M is spherically monotone and the torus action is compressible
– see Remark 1.11.
(23) See [23].
2 Detecting stable displaceability
For detecting stable displaceability of a subset of a symplectic manifold we
shall use the following result (cf. [48, Chapter 6]).
Theorem 2.1. Let X be a closed subset of a closed symplectic manifold
(M,ω). Assume that there exists a contractible loop of Hamiltonian diffeo-
morphisms of (M,ω) generated by a normalized time-periodic Hamiltonian
Ht(x) so that Ht(x) 6= 0 for all t ∈ [0, 1] and x ∈ X. Then X is stably
displaceable.
Proof. Denote by ht the Hamiltonian loop generated by H . Let h
t be its
homotopy to the constant loop: h
t = ht and h
t = 1. Write H
(s)(x, t) for
the corresponding normalized Hamiltonians. Consider the family of diffeo-
morphisms Ψs of M × T
∗S1 given by
Ψs(x, r, θ) = (h
θ x, r −H
(s)(h
θ x, θ), θ) .
One readily checks that Ψs, s ∈ [0, 1], is a Hamiltonian isotopy (not com-
pactly supported). We claim that Ψ1 displaces Y := X × {r = 0}. Indeed,
if Ψ1(x, 0, θ) ∈ Y we have hθx ∈ X and Hθ(hθx) = 0 which contradicts the
assumption of the theorem. This completes the proof.
Proof of Theorem 1.7: Choose a linear functional F : Rk → R with
rational coefficients which is strictly positive on Y . Then for some suffi-
ciently large positive integer N the Hamiltonian H := NΦ∗F generates a
contractible Hamiltonian circle action on M and H is strictly positive on
X := Φ−1(Y ). Thus X is stably displaceable in view of the previous theo-
3 Preliminaries on Hamiltonian Floer theory
3.1 Valuation on QH∗(M)
Define a function ν : K → Γ by
θ) = max{ θ | zθ 6= 0} .
The convention is that ν(0) = −∞. In algebraic terms, exp ν is a non-
Archimedean absolute value on K.
The function ν admits a natural extension to Λ and then to QH∗(M) –
abusing the notation we will denote all of them by ν. Namely, any element
of λ ∈ Λ can be uniquely represented as λ =
θ uθs
θ, where each uθ belongs
to F [q, q−1], and any non-zero a ∈ QH∗(M) can be uniquely represented as
i λibi, 0 6= λi ∈ Λ, 0 6= bi ∈ H∗(M ;F). Define
ν(λ) := max
θ | uθ 6= 0
ν(a) := max
ν(λi).
3.2 Hamiltonian Floer theory
We briefly recall the notation and conventions for the setup of the Hamilto-
nian Floer theory that will be used in the proofs.
Let L be the space of all smooth contractible loops γ : S1 = R/Z → M .
We will view such a γ as a 1-periodic map γ : R → M . Let D2 be the
standard unit disk in R2. Consider a covering L̃ of L whose elements are
equivalence classes of pairs (γ, u), where γ ∈ L, u : D2 → M , u|∂D2 = γ (i.e.
u(e2π
−1t) = γ(t)), is a (piecewise smooth) disk spanning γ in M and the
equivalence relation is defined as follows: (γ1, u1) ∼ (γ2, u2) if and only if γ1 =
γ2 and the 2-sphere u1#(−u2) vanishes in H
2 (M). The equivalence class of
a pair (γ, u) will be denoted by [γ, u]. The group of deck transformations of
the covering L̃ → L can be naturally identified with HS2 (M). An element
A ∈ HS2 (M) acts by the transformation
A([γ, u]) = [γ, u#(−A)]. (8)
Let F :M× [0, 1] → R be a Hamiltonian function (which is time-periodic
as we always assume). Set Ft := F (·, t). We will denote by ft the Hamiltonian
flow generated by F , meaning the flow of the time-dependent Hamiltonian
vector field Xt defined by the formula
ω(·, Xt) = dFt(·) ∀t.
(Note our sign convention!)
Let PF ⊂ L be the set of all contractible 1-periodic orbits of the Hamilto-
nian flow generated by F , i.e. the set of all γ ∈ L such that γ(t) = ft(γ(0)).
Denote by P̃F the full lift of PF to L̃.
Denote by Fix (F ) the set of those fixed points of f that are endpoints of
contractible periodic orbits of the flow:
Fix (F ) := {x ∈M | ∃γ ∈ PF , x = γ(0)}.
We say that F is regular if for any x ∈ Fix (F ) the map dxf : TxM → TxM
does not have eigenvalue 1.
Recall that the action functional is defined on L̃ by the formula:
AF ([γ, u]) =
F (γ(t), t)dt−
Note that
AF (Ay) = AF (y) + ω(A) (9)
for all y ∈ L̃ and A ∈ HS2 (M).
For a regular Hamiltonian F define a vector space C(F ) over F as the
set of all formal sums
λiyi, λi ∈ Λ, yi ∈ P̃F ,
modulo the relations
Ay = s−ω(A)q−c1(A)y,
for all y ∈ P̃F , A ∈ H
2 (M). The grading on Λ together with the Conley-
Zehnder index on elements of P̃F (see Section 3.3) defines a Z-grading on
C(F ). We will denote the i-th graded component by Ci(F ).
Given a loop {Jt}, t ∈ S
1, of ω-compatible almost complex structures,
define a Riemannian metric on L by
(ξ1, ξ2) =
ω(ξ1(t), Jtξ2(t))dt,
where ξ1, ξ2 ∈ TγL. Lift this metric to L̃ and consider the negative gradient
flow of the action functional AF . For a generic choice of the Hamiltonian
F and the loop {Jt} (such a pair (F, J) is called regular) the count of iso-
lated gradient trajectories connecting critical points of AF gives rise in the
standard way [26], [32], [58] to a Morse-type differential
d : C(F ) → C(F ), d2 = 0. (10)
The differential d is Λ-linear and has the graded degree −1. It strictly de-
creases the action. The homology, defined by d, is called the Floer homology
and will be denoted by HF∗(F, J). It is a Λ-module. Different choices of a
regular pair (F, J) lead to natural isomorphisms between the Floer homology
groups.
The following proposition summarizes a few basic algebraic properties of
Floer complexes and Floer homology that will be important for us further.
The proof is straightforward and we omit it.
Proposition 3.1.
1) Each Ci(F ) and each HFi(F, J), i ∈ Z, is a finite-dimensional vector
space over K.
2) Multiplication by q defines isomorphisms Ci(F ) → Ci+2(F ) and
HFi(F, J) → HFi+2(F, J) of K-vector spaces.
3) For each i ∈ Z there exists a basis of Ci(F ) over K consisting of the
elements of the form ql[γ, u], with [γ, u] ∈ P̃F .
4) A finite collection of elements of the form ql[γ, u], [γ, u] ∈ P̃F , lying
in C0(F ) ∪C1(F ) is a basis of the vector space C0(F )⊕C1(F ) over the field
K if and only if it is a basis of the module C(F ) over the ring Λ.
3.3 Conley-Zehnder and Maslov indices
In this section we briefly outline the definition and recall the relevant proper-
ties of the Conley-Zehnder index referring to [54, 58, 57] for details. In par-
ticular, we show that the Conley-Zehnder index is a quasi-morphism on the
universal cover S̃p (2k) of the symplectic group Sp(2k) (see Proposition 3.5
below), a fact which will be useful for asymptotic calculations with Floer
homology in the next sections. There are several routes leading to this fact,
which is quite natural since all homogeneous quasi-morphisms on S̃p (2k) are
proportional, and hence the same quasi-morphism admits quite dissimilar
definitions [7]. We extract the quasi-morphism property from the paper of
Robbin and Salamon [54] by bringing together several statements contained
therein7.
The Conley-Zehnder index assigns to each [γ, u] ∈ P̃F a number. Orig-
inally the Conley-Zehnder index was defined only for regular Hamiltonians
[18] – in this case it is integer-valued and gives rise to a grading of the ho-
mology groups in Floer theory. Later the definition was extended in different
ways by different authors to arbitrary Hamiltonians. We will use such an ex-
tension introduced in [54] (also see [57, 58]). In this case the Conley-Zehnder
index may take also half-integer values.
Let k be a natural number. Consider the symplectic vector space R2k
with a symplectic form ω2k on it. Denote by p = (p1, . . . , pk), q = (q1, . . . , qk)
the corresponding Darboux coordinates on the vector space R2k.
7We thank V.L. Ginzburg for stimulating discussions on the material of this section.
Robbin-Salamon index of Lagrangian paths: Let V ⊂ R2k be a
Lagrangian subspace. Consider the Grassmannian Lagr (k) of all Lagrangian
subspaces in R2k and consider the hypersurface ΣV ⊂ Lagr (k) formed by all
the Lagrangian subspaces that are not transversal to V . To such a V and
to any smooth path {Lt}, 0 ≤ t ≤ 1, in Lagr (k) Robbin and Salamon [54]
associate an index, which may take integer or half-integer values and which
we will denote by RS({Lt}, V ). The definition of the index can be outlined
as follows.
A number t ∈ [0, 1] is called a crossing if Lt ∈ ΣV . To each crossing t one
associates a certain quadratic form Qt on the space L(t) ∩ V – see [54] for
the precise definition. The crossing t is called regular if the quadratic form
Qt is non-degenerate. The index of such a regular crossing t is defined as the
signature of Qt if 0 < t < 1 and as half of the signature of Qt if t = 0, 1.
One can show that regular crossings are isolated. For a path {Lt} with only
regular crossings the index RS({Lt}, V ) is defined as the sum of the indices
of its crossings. An arbitrary path can be perturbed, keeping the endpoints
fixed, into a path with only regular crossings and the index of the perturbed
path does not depend on the perturbation – in fact, it depends only on the
fixed endpoints homotopy class of the path. Moreover, it is additive with
respect to the concatenation of paths and satisfies the naturality property:
RS({ALt}, AV ) = RS({Lt}, V ) for any symplectic matrix A.
Indices of paths in Sp (2k): Consider the group Sp (2k) of symplectic
2k × 2k-matrices. Denote by S̃p (2k) its universal cover. One can use the
index RS in order to define two indices on the space of smooth paths in
Sp (2k).
The first index, denoted by Ind2k, is defined as follows. Fix a Lagrangian
subspace V ⊂ R2k. For each smooth path {At}, 0 ≤ t ≤ 1, in Sp (2k) define
Ind2k ({At}, V ) as
Ind2k ({At}, V ) := RS({AtV }, V ).
The naturality of the RS index implies that
RS({BAtB
−1(BV )}, BV ) = RS({BAtV )}, BV ) =
= RS({AtV )}, V ) for any B ∈ Sp (2k)
and thus we get the following naturality condition for Ind2k:
Ind2k ({BAtB
−1}, BV ) = Ind2k ({At}, V ) for any B ∈ Sp (2k). (11)
The second index, which we will call the Conley-Zehnder index of a matrix
path and which will be denoted by CZmatr, is defined as follows. For each
A ∈ Sp (2k) denote by GrA the graph of A which is a Lagrangian subspace
of the symplectic vector space R4k = R2k×R2k equipped with the symplectic
structure ω4k = −ω2k ⊕ ω2k. Denote by ∆ the diagonal in R
4k = R2k × R2k
– it is a Lagrangian subspace with respect to ω4k. Now for any smooth path
{At}, 0 ≤ t ≤ 1, in Sp (2k) define CZmatr as
CZmatr({At}) := RS({GrAt},∆).
Equivalently, one can define CZmatr({At}) similarly to the index RS by look-
ing at the intersections of {A(t)} with the hypersurface Σ ⊂ Sp (2k) formed
by all the symplectic 2k× 2k-matrices with eigenvalue 1 and translating the
notions of a regular crossing and the corresponding quadratic form to this
setup.
Both indices Ind2k ({At}, V ) and CZmatr({At}) depend only on the fixed
endpoints homotopy class of the path {At} and are additive with respect to
the concatenation of paths in Sp (2k). The relation between the two indices
is as follows. Denote by I2k the 2k × 2k identity matrix. Given a smooth
path {At}, 0 ≤ t ≤ 1, in Sp (2k), set Ât := I2k ⊕ At ∈ Sp (4k). Then
CZmatr({At}) = Ind4k({Ât},∆). (12)
Remark 3.2. Note that near each W ∈ ΣV there exists a local coordinate
chart (on Lagr (k)) in which ΣV can be defined by an algebraic equation of
degree bounded from above by a constant C depending only on k and W .
Moreover, since for any two V, V ′ ∈ Lagr (k) there exists a diffeomorphism of
Lagr (k) mapping ΣV into ΣV ′ we can assume that C = C(k) is independent
of W and depends only on k. Therefore for any V , for any point W ∈ ΣV
and for any sufficiently small open neighborhood UW of W in Lagr (k) the
number of connected components of UW \(UW ∩ΣV ) is bounded by a constant
depending only on k.
Using these observations and the fact that regular crossings are isolated it
is easy to show that there exists a constant C(k), depending only on k, such
that for any Lagrangian subspace V ⊂ R2k and any path {At} ⊂ Sp (2k),
0 ≤ t ≤ 1, there exists a δ > 0 such that for any smooth path {A′t} ⊂ Sp (2k),
0 ≤ t ≤ 1, which is δ-close to {At} in the C
0-metric, one has
|Ind2k({At}, V )− Ind2k({A
t}, V | < C(k),
|CZmatr({At})− CZmatr({A
t}| < C(k).
Leray theorem on the index Ind2k: The following result follows from
Theorem 5.1 in [54] which Robbin and Salamon credit to Leray [35], p.52.
Denote by L the Lagrangian (q1, . . . , qk)-coordinate plane in R
2k. Any sym-
plectic matrix S ∈ Sp (2k) can be decomposed into k × k blocks as
where the blocks satisfy, in particular, the condition that
EF T − FET = 0. (13)
If SL ∩ L = 0 then the k × k-matrix F is invertible and multiplying (13)
by F−1 on the left and (F T )−1 = (F−1)T on the right, we get that F−1E −
ET (F−1)T = 0. Therefore the matrix QS := F
−1E is symmetric.
Theorem 3.3 ([54], Theorem 5.1; [35], p.52). Assume {At}, {Bt}, 0 ≤ t ≤ 1,
are two smooth paths in Sp (2k), such that A0 = B0 = I2k and A1L ∩ L = 0,
B1L ∩ L = 0, A1B1L ∩ L = 0. Then
Ind2k({AtBt}, L) = Ind2k({At}, L) + Ind2k({Bt}, L) +
sign (QA1 +QB1),
where sign (QA1 +QB1) is the signature of the quadratic form defined by the
symmetric k × k-matrix QA1 +QB1.
Corollary 3.4. Let V be any Lagrangian subspace of R2k. Then there exists
a positive constant C, depending only on k, such that for any smooth paths
{Xt}, {Yt}, 0 ≤ t ≤ 1, in Sp (2k), such that X0 = Y0 = I2k (there are no
assumptions on X1, Y1!),
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| < C.
Proof. We will write C1, C2, . . . for (possibly different) positive constants de-
pending only on k.
Pick a map Ψ ∈ Sp (2k) such that ΨV = L. Denote At = ΨXtΨ
Bt = ΨYtΨ
−1. Note that the paths {At}, {Bt} are based at the identity.
Using the naturality property (11) of Ind2k we get
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| =
= |Ind2k({ΨXtYtΨ
−1},ΨV )− Ind2k({ΨXtΨ
−1},ΨV )−
−Ind2k({ΨYtΨ
−1},ΨV )| =
= |Ind2k({(ΨXtΨ
−1)(ΨYtΨ
−1)}, L)− Ind2k({ΨXtΨ
−1}, L)−
−Ind2k({ΨYtΨ
−1}, L)| =
= |Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|.
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| =
= |Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|. (14)
Further on, Remark 3.2 implies that we can find sufficiently C0-close identity-
based perturbations {A′t}, {B
t} of {At}, {Bt} such that
A′1L ∩ L = 0, B
1L ∩ L = 0, A
1L ∩ L = 0. (15)
|Ind2k({AtBt}, L)− Ind2k({At}, L)− Ind2k({Bt}, L)|−
−|Ind2k({A
t}, L)− Ind2k({A
t}, L)− Ind2k({B
t}, L)| < C1, (16)
for some C1. On the other hand, since the three identity-based paths {A
{B′t}, {A
t}, satisfy the conditions (15), we can apply to them Theorem 3.3.
Hence there exists C2 such that
|Ind2k({A
t}, L)− Ind2k({A
t}, L)− Ind2k({B
t}, L)| < C2.
Combining it with (14) and (16) we get that there exists C3 such that
|Ind2k({XtYt}, V )− Ind2k({Xt}, V )− Ind2k({Yt}, V )| < C3,
which finishes the proof.
Conley-Zehnder index as a quasi-morphism: Recall that 2n = dimM .
Restricting CZmatr to the identity-based paths in Sp (2n) one gets a function
on S̃p (2n) that will be still denoted by CZmatr.
Proposition 3.5 (cf. [19]). The function CZmatr : S̃p (2n) → R is a quasi-
morphism. It means that there exists a constant C > 0 such that
|CZmatr(ab)− CZmatr(a)− CZmatr(b)| ≤ C ∀a, b ∈ S̃p (2n).
Proof. Represent a and b by identity-based paths {At}, {Bt}, 0 ≤ t ≤ 1, in
Sp (2n). Then use (12) and apply Corollary 3.4 for k = 2n, V = ∆ to {Ât},
{B̂t} in Sp (4n).
Maslov index of symplectic loops: The Conley-Zehnder index for
identity-based loops in Sp (2n) is called the Maslov index of a loop. Its
original definition, going back to [4], is the following: it is the intersection
number of an identity-based loop with the stratified hypersurface Σ whose
principal stratum is equipped with a certain co-orientation. Note that we do
not divide the intersection number by 2 and thus in our case the Maslov index
takes only even values; for instance, the Maslov index of a counterclockwise
2π-twist of the standard symplectic R2 is 2. We denote the Maslov index of
a loop {B(t)} by Maslov ({B(t)}).
Conley-Zehnder and Maslov indices of periodic orbits: The Con-
ley-Zehnder index for periodic orbits is defined by means of the Conley-
Zehnder index for matrix paths as follows. Given [γ, u] ∈ P̃F , build an
identity-based path {A(t)} in Sp (2n) as follows: take a symplectic trivial-
ization of the bundle u∗(TM) over D2 and use the trivialization to identify
the linearized flow dγ(0)ft, 0 ≤ t ≤ 1, along γ with a symplectic matrix
{A(t)}. Then the Conley-Zehnder index CZF ([γ, u]) is defined as
CZF ([γ, u]) := n− CZmatr ({A(t)}). (17)
With such a normalization of CZF for any sufficiently C
2-small autonomous
Morse Hamiltonian F , the Conley-Zehnder index of an element of P̃F , rep-
resented by a pair [x, u] consisting of a critical point x of F (viewed as a
constant path in M) and the trivial disk u, is equal to the Morse index of
x. Note that with such a normalization CZF (Sy) = CZF (y)+2
c1(M) for
every y ∈ P̃F and S ∈ H
2 (M).
Similarly, if the time-1 flow generated by F defines a loop in Ham(M) then
to each [γ, u] ∈ P̃F one can associate its Maslov index. Namely, trivialize the
bundle u∗(TM) over D2 and identify the linearized flow {dxft} along γ with
an identity-based loop of symplectic 2n × 2n-matrices. Define the Maslov
index mF ([γ, u]) as the Maslov index for the loop of symplectic matrices.
Under the action of HS2 (M) on P̃F the Maslov index changes as follows:
mF (S · [γ, u]) = mF ([γ, u])− 2
c1(M), S ∈ H
2 (M).
Let us make the following remark. Assume γ ∈ PF and assume that a
symplectic trivialization of the bundle γ∗(TM) over S1 identifies {dγ(0)ft}
with an identity-based path {A(t)} of symplectic matrices. Assume there
is another symplectic trivialization of the same bundle, coinciding with the
first one at γ(0), and denote by {B(t)} the identity-based loop of transition
matrices from the first symplectic trivialization to the second one. Use the
second trivialization to identify {dγ(0)ft} with an identity-based path {A
′(t)}.
CZmatr ({A
′(t)}) = CZmatr ({A(t)}) +Maslov ({B(t)}), (18)
and if {A(t)} is a loop then so is {A′(t)} and
Maslov ({A′(t)}) = Maslov ({A(t)}) +Maslov ({B(t)}). (19)
3.4 Spectral numbers
Given the algebraic setup as above, the construction of the Piunikhin-Sala-
mon-Schwarz (PSS) isomorphism [47] yields a Λ-linear isomorphism (PSS-
isomorphism) φM : QH∗(M) → HF∗(F, J) which preserves the grading and
which is actually a ring isomorphism (the pair-of-pants product defines a ring
structure on HF∗(F, J)).
Using the PSS-isomorphism one defines the spectral numbers c(a, F ),
where 0 6= a ∈ QH∗(M), in the usual way [45]. Namely, the action functional
AF defines a filtration on C(F ) which induces a filtration HF
∗ (F, J), α ∈ R,
on HF∗(F, J), with HF
∗ (F, J) ⊂ HF
∗ (F, J) as long as α < β. Then
c(a, F ) := inf {α | φM(a) ∈ HF
∗ (F, J)}.
Such spectral number is finite and well-defined (does not depend on J). Here
is a brief account of the relevant properties of spectral numbers – for details
see [45] (see also [65, 42, 59, 43] for earlier versions of this theory).
(Spectrality) c(a,H) ∈ spec (H), where the spectrum spec (H) of H is
defined as the set of critical values of the action functional AH , i.e.
spec (H) := AH(P̃H) ⊂ R.
(Quantum homology shift property) c(λa,H) = c(a,H) + ν(λ) for all
λ ∈ Λ, where ν is the valuation defined in Section 3.1.
(Hamiltonian shift property) c(a,H + λ(t)) = c(a,H) +
λ(t) dt for
any Hamiltonian H and function λ : S1 → R.
(Monotonicity) If H1 ≤ H2, then c(a,H1) ≤ c(a,H2).
(Lipschitz property) The map H 7→ c(a,H) is Lipschitz on the space of
(time-dependent) Hamiltonians H : M × S1 → R with respect to the
C0-norm.
(Symplectic invariance) c(a, φ∗H) = c(a,H) for every φ ∈ Symp0(M),
H ∈ C∞(M); more generally, Symp (M) acts on H∗(M ;F), and hence
on QH∗(M), and c(a, φ
∗H) = c(φ∗a,H) for any φ ∈ Symp (M).
(Normalization) c(a, 0) = ν(a) for every a ∈ QH∗(M).
(Homotopy invariance) c(a,H1) = c(a,H2) for any normalized H1, H2
generating the same φ ∈ H̃am (M). Thus one can define c(a, φ) for any
φ ∈ H̃am (M) as c(a,H) for any normalized H generating φ.
(Triangle inequality) c(a ∗ b, φψ) ≤ c(a, φ) + c(b, ψ).
The commutative ring QH•(M) admits a K-bilinear and K-valued form Ω
on QH•(M) which associates to a pair of quantum homology classes a, b ∈
QH•(M) the coefficient (belonging to K) at the class [point] = [point] · q
a point in their quantum product a ∗ b ∈ QH•(M) (the Frobenius structure).
Let τ : K → F be the map sending each series
θ∈Γ zθs
θ, zθ ∈ F , to its free
term z0. Define a non-degenerate F -valued F -linear pairing on QH•(M) by
Π(a, b) := τΩ(a, b) = τΩ(a ∗ b, [M ]) . (20)
Note that Π is symmetric and
Π(a ∗ b, c) = Π(a, b ∗ c) ∀a, b, c ∈ QH•(M). (21)
With this notion at hand, we can present another important property of
spectral numbers:
(Poincaré duality) c(b, φ) = − infa∈Υ(b) c(a, φ
−1) for all b ∈ QH•(M)\{0}
and φ. Here Υ(b) denotes the set of all a ∈ QH•(M) with Π(a, b) 6= 0.
The Poincaré duality can be extracted from [47] (cf. [22]) – for a proof see
[46].
The next property is an immediate consequence of the definitions (see [22]
for a discussion in the monotone case):
(Characteristic exponent property) Given 0 6= λ ∈ F , a, b ∈ QH∗(M),
a, b, a + b 6= 0, and a (time-dependent) Hamiltonian H , one has
c(λ · a,H) = c(a,H) and c(a+ b,H) ≤ max(c(a,H), c(b,H)).
3.5 Partial symplectic quasi-states
Given a non-zero idempotent a ∈ QH2n(M) and a time-independent Hamil-
tonian H :M → R, define
ζ(a,H) := lim
c(a, lH)
. (22)
When a is fixed, we shall often abbreviate ζ(H) instead of ζ(a,H). The limit
in the formula (22) always exists and thus the functional ζ : C∞(M) → R is
well-defined. The functional ζ on C∞(M) is Lipschitz with respect to the C0-
norm ‖H‖ = maxM |H| and therefore extends to a functional ζ : C(M) → R,
where C(M) is the space of all continuous functions on M . These facts were
proved in [23] in the case a = [M ] but the proofs actually go through for any
non-zero idempotent a ∈ QH2n(M).
Here we will list the properties of ζ for such an M . Again, these proper-
ties were proved in [23] in the case a = [M ] but the proof goes through for
any non-zero idempotent a ∈ QH2n(M). The additivity with respect to con-
stants property was not explicitly listed in [23] but follows immediately from
the definition of ζ and the Hamiltonian shift property of spectral numbers.
The triangle inequality follows readily from the definition of ζ and from the
triangle inequality for the spectral numbers.
Theorem 3.6. The functional ζ : C(M) → R satisfies the following prop-
erties:
Semi-homogeneity: ζ(αF ) = αζ(F ) for any F and any α ∈ R≥0.
Triangle inequality: If F1, F2 ∈ C
∞(M), {F1, F2} = 0 then ζ(F1 + F2) ≤
ζ(F1) + ζ(F2).
Partial additivity and vanishing: If F1, F2 ∈ C
∞(M), {F1, F2} = 0 and the
support of F2 is displaceable, then ζ(F1 + F2) = ζ(F1); in particular, if the
support of F ∈ C(M) is displaceable, ζ(F ) = 0.
Additivity with respect to constants and normalization: ζ(F +α) = ζ(F )+α
for any F and any α ∈ R. In particular, ζ(1) = 1.
Monotonicity: ζ(F ) ≤ ζ(G) for F ≤ G.
Symplectic invariance: ζ(F ) = ζ(F ◦ f) for every symplectic diffeomorphism
f ∈ Symp0 (M).
Characteristic exponent property: ζ(a1+a2, F ) ≤ max(ζ(a1, F ), ζ(a2, F )) for
each pair of non-zero idempotents a1, a2 with a1 ∗ a2 = 0, a1+ a2 6= 0 (in this
case a1 + a2 is also a non-zero idempotent), and for all F ∈ C(M) .
We will call the functional ζ : C(M) → R satisfying all the properties
listed in Theorem 3.6 a partial symplectic quasi-state.
4 Basic properties of (super)heavy sets
In this section we prove parts (i) and (iii) of Theorem 1.2, as well as The-
orem 1.3. We shall use that a partial symplectic quasi-state ζ extends by
continuity in the uniform norm to a monotone functional on the space of
continuous functions C(M), see Section 3.5 above. In particular, one can
use continuous functions instead of the smooth ones in the definition of (su-
per)heaviness in formulae (3) and (4).
Assume a partial quasi-state ζ defined by a non-zero idempotent is fixed
and we consider heaviness and superheaviness with respect to ζ . We start
with the following elementary
Proposition 4.1. A closed subset X ⊂ M is heavy if and only if for every
H ∈ C∞(M) with H|X = 0, H ≤ 0 one has ζ(H) = 0. A closed subset
X ⊂ M is superheavy if and only if for every H ∈ C∞(M) with H|X = 0,
H ≥ 0 one has ζ(H) = 0.
Proof. The “only if” parts follow readily from the monotonicity property of
ζ . Let us prove the “if” part in the “heavy case” – the “superheavy” case is
similar. Take a function H on M and put
F = min(H − inf
H, 0) .
Note that F |X = 0 and F ≤ 0. Thus ζ(F ) = 0 by the assumption of the
proposition. Thus
0 = ζ(F ) ≤ ζ(H − inf
H) = ζ(H)− inf
which yields heaviness of X .
The following proposition proves part (i) of Theorem 1.2.
Proposition 4.2. Every superheavy set is heavy.
Proof. Let X ⊂ M be a superheavy subset. Assume that H|X = 0, H ≤ 0.
By the triangle inequality for ζ we have ζ(H) + ζ(−H) ≥ 0. Note that
−H|X = 0, −H ≥ 0. Superheaviness yields ζ(−H) = 0, so ζ(H) ≥ 0. But
by monotonicity ζ(H) ≤ 0. Thus ζ(H) = 0 and the claim follows from
Proposition 4.1.
Superheavy sets have the following user-friendly property.
Proposition 4.3. Let X ⊂ M be a superheavy set. Then for every α ∈ R
and H ∈ C∞(M) with H|X ≡ α one has ζ(H) = α.
Proof. Since ζ(H + α) = ζ(H) + α it suffices to prove the proposition for
α = 0. Take any function H with H|X = 0. Since X is superheavy and, by
Proposition 4.2, also heavy, we have
0 = ζ(−|H|) ≤ ζ(H) ≤ ζ(|H|) = 0 ,
which yields ζ(H) = 0.
As an immediate consequence we get part (iii) of Theorem 1.2.
Proposition 4.4. Every superheavy set intersects with every heavy set.
Proof. Let X be a superheavy set and Y be a heavy set. Assume on the
contrary that X ∩ Y = ∅. Take a function H ≤ 0 with H|Y ≡ 0 and
H|X ≡ −1. Then ζ(H) = −1 by Proposition 4.3. On the other hand,
ζ(H) = 0 since Y is heavy, and we get a contradiction.
Note that two heavy sets do not necessarily intersect each other: a meridian
of T2 is heavy (see Corollary 6.4 below), while two meridians can be disjoint.
Proof of Theorem 1.3 (i) and (ii): The triangle inequality yields
c(a,H) = c(a ∗ [M ], 0 +H) ≤ c(a, 0) + c([M ], H) = ν(a) + c([M ], H).
Passing to the partial quasi-states ζ(a,H) and ζ([M ], H) we get:
ζ(a,H) = lim
c(a, kH)/k ≤
≤ lim
(ν(a) + c([M ], kH))/k = lim
c([M ], kH)/k = ζ([M ], H).
The result now follows from the definition of heavy and superheavy sets (see
Definition 1.1).
Proof of Theorem 1.3 (iii): By the characteristic exponent property of
spectral invariants,
ζ(a, F ) ≤ max
i=1,...,l
ζ(ei, F ) ∀F ∈ C
∞(M) . (23)
Choose a sequence of functions Gj ∈ C
∞(M), j → +∞, with the fol-
lowing properties: Gk ≤ Gj for k > j, Gj = 0 on X , Gj ≤ 0 and for
every function F ≤ 0 which vanishes on an open neighborhood of X there
exists j so that Gj ≤ F (existence of such a sequence can be checked easily).
In view of inequality (23), we have that for every j there exists i so that
ζ(a,Gj) ≤ ζ(ei, Gj). Passing, if necessary, to a subsequence Gjk , jk → +∞,
we can assume without loss of generality that i is the same for all j. In view
of heaviness of X with respect to a, we have that ζ(a,Gj) = 0. Therefore
ζ(ei, Gj) ≥ 0.
Choose any function F ≤ 0 onM which vanishes on an open neighborhood
of X . Then there exists j large enough so that F ≥ Gj. By monotonicity
combined with the previous estimate we have
0 ≥ ζ(ei, F ) ≥ ζ(ei, Gj) ≥ 0 ,
which yields ζ(ei, F ) = 0.
Now let F be any continuous function on M that vanishes on X . Take
a sequence of continuous functions Fj , converging to F in the C
0-norm, so
that each Fj vanishes on an open neighborhood of X . Then ζ(ei, Fj) =
limj→+∞ ζ(ei, Fj) = 0, because ζ(ei, ·) is Lipschitz with respect to the C
norm. The heaviness ofX with respect to ei now follows from Proposition 4.1.
This finishes the proof of the theorem.
5 Products of (super)heavy sets
In this section we prove Theorem 1.5 on products of (super)heavy subsets.
5.1 Product formula for spectral invariants
The proof of Theorem 1.5 is based on the following general result.
Theorem 5.1. For every pair of time-dependent Hamiltonians G1, G2 onM1
and M2, and all non-zero a1 ∈ QHi1(M1), a2 ∈ QHi2(M2) we have
c(a1 ⊗ a2, G1(z1, t) + G2(z2, t)) = c(a1, G1) + c(a1, G2) .
Here G1(z1, t) +G2(z2, t) is a time-dependent Hamiltonian on M1 ×M2.
Let us deduce Theorem 1.5 from Theorem 5.1.
Proof of Theorem 1.5: We show that the product of superheavy sets is
superheavy (the proof for heavy sets goes without any changes). We denote
by ζ1, ζ2 and ζ the partial quasi-states on M1,M2 and M := M1 ×M2 as-
sociated to the idempotents a1, a2 and a1 ⊗ a2 respectively. Let Xi ⊂ Mi,
i = 1, 2, be a superheavy set. By Proposition 4.1 it suffices to show that if a
non-negative function G ∈ C∞(M) vanishes on some neighborhood, say U ,
of X := X1 × X2 then ζ(G) = 0. (Since ζ is Lipschitz with respect to the
C0-norm this would imply that ζ(G) = 0 for any non-negative G ∈ C(M)
that vanishes on X). Put K := maxM G. Choose neighborhoods Ui of Xi so
that U1 ×U2 ⊂ U . Choose non-negative functions Gi on Mi which vanish on
Xi and such that Gi(z) > K for all z ∈Mi \Ui. Observe that G ≤ G1 +G2.
But, in view of Theorem 5.1 and superheaviness of Xi, we have
ζ(G1 +G2) = ζ1(G1) + ζ2(G2) = 0 .
By monotonicity
0 ≤ ζ(G) ≤ ζ(G1 +G2) = 0 ,
and thus ζ(G) = 0.
It remains to prove Theorem 5.1. Note that the left-hand side of the equality
stated in the theorem does not exceed the right-hand side: this is an imme-
diate consequence of the triangle inequality for spectral invariants. However,
we were unable to use this observation for proving the theorem. Our ap-
proach is based on a rather lengthy algebraic analysis which enables us to
calculate separately the left and the right-hand sides “on the chain level”. A
simple inspection of the results of this calculation yields the desired equality.
5.2 Decorated Z2-graded complexes
A Z2-complex is a Z2-graded finite-dimensional vector space V over a field
K equipped with a K-linear differential ∂ : V → V satisfying ∂2 = 0 and
shifting the grading. A decorated complex over K = KΓ includes the following
data:
• a countable subgroup Γ ⊂ R;
• a Z2-graded complex (V, d) over KΓ;
• a preferred basis x1, . . . , xn of V ;
• a function F : {x1, . . . , xn} → R (called the filter) which extends to V
λjxj) = max{ν(λj) + F (xj)
∣∣∣ λj 6= 0},
and satisfies F (dv) < F (v) for all v ∈ V \ {0}. The convention is that
F (0) = −∞. Here ν is the valuation defined in Section 3.1 above.
We shall use the notation
V := (V, {xi}i=1,...,n, F, d,Γ)
for a decorated complex.
The ⊗̂K-tensor product V = V1⊗̂KV2 of decorated complexes
Vi = (Vi, {x
j }j=1,...,ni, Fi, di,Γi) , i = 1, 2
is defined as follows. Consider the space V = V1⊗̂KV2 (see formula (5) above)
with the natural Z2-grading. Define the differential d on V by
d(x⊗ y) = d1x⊗ y + (−1)
deg xx⊗ d2y .
The preferred basis in V is given by {xpq := x
p ⊗ x
q } and the filter F is
defined by
F (xpq) = F1(x
p ) + F2(x
Finally, we put V := (V, {xpq}, F, d,Γ1 + Γ2) .
The (Z2-graded) homology of decorated complexes are denoted by H∗(V)
– they are K-vector spaces. By the Künneth formula, H(V1⊗̂KV2) =
H(V1)⊗̂KH(V2).
Next we define spectral invariants associated to a decorated complex V :=
(V, {xpq}, F, d) . Namely, for a ∈ H(V) put
c(a) := inf{F (v) | a = [v], v ∈ Ker d} .
We shall see below that c(a) > −∞ for each a 6= 0.
The purpose of this algebraic digression is to state the following result:
Theorem 5.2. For any two decorated complexes V1,V2
c(a1 ⊗ a2) = c(a1) + c(a2) ∀a1 ∈ H(V1), a2 ∈ H(V2)
5.3 Reduced Floer and Quantum homology
The 2-periodicity of the Floer complex and Floer homology defined by the
multiplication by q (see Proposition 3.1 above) allows to encode their al-
gebraic structure in a decorated Z2-complex. Consider a regular pair (G, J)
consisting of a Hamiltonian function and a compatible almost-complex struc-
ture on M (both, in general, are time-dependent). Let (C∗(G), dG,J) be the
corresponding Floer complex. Let us associate to it a Z2-complex: a Z2-
graded vector space VG over KΓ, defined as
VG := C0(G)⊕ C1(G),
with the obvious Z2-grading, and a differential ∂G,J : VG → VG, defined as
the direct sum of dG,J : C1(G) → C0(G) and qdG,J : C0(G) → C1(G). One
readily checks that this is indeed a Z2-complex because dG,J : C(G) → C(G)
is ΛΓ-linear. We will call (VG, ∂G,J) the Z2-complex associated to (G, J).
Note that the cycles and the boundaries of (VG, ∂G) having Z2-degree
i ∈ {0, 1} in VG coincide, respectively, with the cycles and the boundaries
having Z-degree i of (C(G), dG,J). Therefore the Floer homology HFi(G, J)
is isomorphic, as a vector space over KΓ, to the i-th degree component of the
homology of the complex (VG, ∂G,J).
The Z2-complex (VG, ∂G,J) carries a structure of the decorated complex
VG,J as follows. Let γi(t), i = 1, . . . , m, be the collection of all contractible
1-periodic orbits of the Hamiltonian flow generated by G. Choose disc ui
in M spanning γi. For each i there exists unique integer, say ri, so that
the Conley-Zehnder index of the element xi := q
ri · [γi, ui] lies in the set
{0, 1}. Clearly, the collection {xi} forms a basis of VG over KΓ. We shall
consider it as a preferred basis. Note that the preferred basis is unique up to
multiplication of xi’s by elements of the form s
αi , αi ∈ Γ. Finally, the action
functional associated to G defines a filtration on VG.
The homology of (VG, ∂G,J) can be canonically identified via the PSS-
isomorphism with the object which we call reduced quantum homology:
QHred(M) := QH0(M)⊕QH1(M) .
We call this isomorphism the reduced PSS-isomorphism and denote it by ψG,J .
Note that we have a natural projection p : QH∗(M) → QHred(M) which
sends any degree homogeneous element a to aqr with deg a + 2r ∈ {0, 1}.
With this notation, the usual Floer-homological spectral invariant c(a,G)
coincides with the spectral invariant c(p(a)) of the decorated complex VG,J .
5.4 Proof of Theorem 5.1
By the Lipschitz property of spectral numbers it is enough to consider the
case when G1 and G2 belong to regular pairs (Gi, Ji), i = 1, 2. Set
G(z1, z2, t) := G1(z1, t) +G(z2, t)
and J := J1 × J2. Then (G, J) is also a regular pair. Put Γi = Γ(Mi, ωi). It
is straightforward to see that the decorated complex VG,J is the ⊗̂K-tensor
product of the decorated complexes VGi,Ji for i = 1, 2.
Put (M,ω) = (M1×M2, ω1⊕ω2). An obvious modification of the Künneth
formula for quantum homology (see e.g. [41, Exercise 11.1.15] for the state-
ment in the monotone case) yields a natural monomorphism
ı : QHi1(M1, ω1)⊗̂KQHi2(M1, ω1) → QHi1+i2(M,ω) .
Since in our setting quantum homologies are 2-periodic, the collection of these
isomorphisms for all pairs (i1, i2) from the set {0, 1} induces an isomorphism
j : QHred(M1)⊗̂KQHred(M2) → QHred(M) .
It has the following properties: First, given two elements a1 ∈ QHi1(M1, ω1)
and a2 ∈ QHi2(M2, ω2) we have that
p(a1)⊗ p(a2) = p(a1 ⊗ a2) .
Second, the following diagram commutes:
H(VG1, ∂G1,J1)⊗̂KH(VG2 , ∂G2,J2)
ψG1,J1⊗ψG2,J2
H(VG, ∂G,J)
QHred(M1)⊗̂KQHred(M2)
// QHred(M)
Here k is the isomorphism coming from the Künneth formula for Z2-comple-
xes, and ψGi,Ji, ψG,J stand for the reduced PSS-isomorphisms. It follows that
the definition of c(ai, Gi), c(a1⊗a2, G) matches the definition of c(p(ai)) and
c(p(a1)⊗ p(a2)). By Theorem 5.2 we get that
c(a1⊗a2, G) = c(p(a1)⊗p(a2)) = c(p(a1))+ c(p(a2)) = c(a1, G1)+ c(a2, G2) .
This proves Theorem 5.1 modulo Theorem 5.2.
5.5 Proof of algebraic Theorem 5.2
A decorated complex is called generic if F (xi) − F (xj) /∈ Γ for all i 6= j
(recall that under our assumptions Γ, the group of periods of the symplectic
form ω over π2(M), is a countable subgroup of R). We start from some
auxiliary facts from linear algebra. Let V := (V, {xi}i=1,...,n, F, d,Γ) be a
generic decorated complex. We recall once again that for brevity we write K
instead of KΓ wherever it is clear what Γ is taken.
An element x ∈ V is called normalized if
x = xp +
i 6=p
λixi , λi ∈ K, F (xp) > max
i 6=p
F (λixi) .
We shall use the notation x = xp+o(xp). In generic complexes, every element
x 6= 0 can be uniquely written as x = λ(xp+o(xp)) for some p = 1, . . . , n and
λ ∈ K. A system of vectors e1, . . . , em in V is called normal if every ei has
the form ei = xji +o(xji) for ji ∈ {1, . . . , n} and the numbers ji are pair-wise
distinct.
Lemma 5.3. Let e1, . . . , em be a normal system. Then
λiei) = max
F (λiei) .
Proof. We prove the result using induction in m. For m = 1 the statement is
obvious. Let’s check the induction step m− 1 → m. Observe that it suffices
to check that
F (e1 +
λiei) ≥ F (e1) . (24)
Then obviously
λiei) ≥ max
F (λiei) ,
while the reversed inequality is an immediate consequence of the definitions.
By the induction step,
λiei) = max
i=2,...,n
F (λiei) .
In view of the genericity, the maximum at the right hand side can be uniquely
written as F (λi0xi0). Without loss of generality we shall assume that ei =
xi + o(xi) and i0 = 2.
λ−12 λiei = x2 + o(x2) .
Write
e1 = x1 + αx2 +X, v = x2 + βx1 + Y,
where α, β ∈ K and X, Y ∈ SpanK(x3, . . . , xn). Note that F (x1) > F (αx2),
F (x2) > F (βx1), which yields
ν(α) < F (x1)− F (x2) < −ν(β) = ν(β
−1) . (25)
In particular, ν(α) < ν(β−1). Note that
e1 + λ2v = (1 + λ2β)x1 + (α + λ2)x2 + Z, Z ∈ SpanK(x3, . . . , xn) .
F (e1 + λ2v) ≥ max(ν(1 + λ2β) + F (x1), ν(α + λ2) + F (x2)) .
If ν(1 + λ2β) ≥ 0 we have F (e1 + λ2v) ≥ F (x1) = F (e1) and inequality (24)
follows. Assume that ν(1+λ2β) < 0 = ν(1). Then ν(λ2β) = 0 = ν(λ2)+ν(β),
and hence ν(λ2) = ν(β
−1) 6= ν(α). Thus
ν(α + λ2) ≥ ν(λ2) = −ν(β) .
Combining this inequality with (25) we get that
F (e1 + λ2v) ≥ ν(α + λ2) + F (x1) + (F (x2)− F (x1))
≥ F (x1) + (ν(α + λ2) + ν(β)) ≥ F (x1) = F (e1) .
This completes the proof of inequality (24), and hence of the lemma.
It readily follows from the lemma that every normal system is linearly inde-
pendent.
Lemma 5.4. Every subspace L ⊂ V has a normal basis.
Proof. We use induction over m = dimK L. The case m = 1 is obvious,
so let us handle the induction step m − 1 → m. It suffices to show the
following: Let e1, . . . , em−1 be a normal basis in a subspace L
′, and let v /∈ L′
be any vector. Put L = SpanK(L
′ ∪ {v}). Then there exists em ∈ L so that
e1, . . . , em is a normal basis. Indeed, assume without loss of generality that
for all i = 1, . . . , m−1 one has ei = xi+ o(xi). Put W = SpanK(xm, . . . , xn).
We claim that L′ ∩W = {0}. Indeed, otherwise
λ1e1 + . . .+ λm−1em−1 = λmxm + . . .+ λnxn
where the linear combinations in the right and the left-hand sides are non-
trivial. Apply F to both sides of this equality. By Lemma 5.3
F (λ1e1 + . . .+ λm−1em−1) = F (xp) mod Γ, where 1 ≤ p ≤ m− 1 ,
while
F (λmxm + . . .+ λnxn) = F (xq) mod Γ, where q ≥ m .
This contradicts the genericity of our decorated complex, and the claim fol-
lows. Since dimL′+dimW = dimV , we have that V = L′⊕W . Decompose v
as u+w with u ∈ L′, w ∈ W , and note that w ∈ L. Note that e1, . . . , em−1, w
are linearly independent. Furthermore, w = λ(xp + o(xp)) for some p ≥ m.
Put em = λ
−1w. The vectors e1, . . . , em form a normal basis in L.
The same proof shows that if L1 ⊂ L2 are subspaces of V , every normal basis
in L1 extends to a normal basis in L2.
Now we turn to the analysis of the differential d. Choose a normal basis
g1, . . . , gq in Im d, and extend it to a normal basis g1, . . . , gq, h1, . . . , hp in
Ker d. Note that each of these p + q vectors has the form xj + o(xj) with
distinct j. Let us assume without loss of generality that the remaining n−p−q
elements of the preferred basis in V are x1, . . . , xq, and
gi = xi+q + o(xi+q), hj = xj+2q + o(xj+2q) .
Here we use that, by the dimension theorem, n = p+ 2q. Note that
x1, . . . , xq, g1, . . . , gq, h1, . . . , hp
is a normal system, and hence a basis in V . We call such a basis a spectral
basis of the decorated complex V.
Note that [h1], . . . , [hp] is a basis in the homology H(V). Consider any
homology class a =
λi[hi]. Every element v ∈ V with a = [v] can be
written as v =
λihi +
αjgj. Thus, by Lemma 5.3, F (v) ≥ maxi F (λihi)
and hence
c(a) = max
F (λihi) . (26)
This proves in particular that the spectral invariants are finite provided a 6= 0.
For finite sets A = {v1, . . . , vs} and B = {w1, . . . , ws} we write A⊗B for the
finite set {vi ⊗ wj}.
Assume now that V1,V2 are generic decorated complexes. We say that they
are in general position if their tensor product V = V1⊗̂KV2 is generic. Let
Bi = {x
1 , . . . , x
1 , . . . , g
1 , . . . , h
}, i = 1, 2
be a spectral basis in Vi. Obviously, B1 ⊗ B2 is a normal basis in V1⊗̂KV2.
We shall denote by d1, d2, d the differentials and by F1, F2, F the filters in
V1,V2 and V respectively. Put Gi = {g
1 , . . . , g
qi }, Hi = {h
1 , . . . , h
and K = G1 ⊗ B2 ∪B1 ⊗G2. Observe that
Im d ⊂W := Span(K) .
Take any two classes
j ] ∈ H(Vi) , i = 1, 2.
Suppose that a1 ⊗ a2 = [v]. Then v is of the form
λ(1)m λ
m ⊗ h
l + w
where w must lie in W . Observe that (H1 ⊗H2) ∩K = ∅. By Lemma 5.3,
F (v) ≥ max
F (λ(1)m λ
m ⊗ h
l ) ,
and hence
c(a1 ⊗ a2) = max
F (λ(1)m λ
m ⊗ h
= max
m ) + F2(λ
= max
m ) + max
l ) = c(a1) + c(a2) .
In the last equality we used (26). This completes the proof of Theorem 5.2
for decorated complexes in general position.
It remains to remove the general position assumption. This will be done
with the help of the following lemma. We shall work with a family of deco-
rated complexes
V := (V, {xi}i=1,...,n, F, d,Γ)
which have exactly the same data (preferred basis, grading, differential and
Γ) with the exception of the filter F which will be allowed to vary in the class
of filters. The corresponding spectral invariants will be denoted by c(a, F ).
Lemma 5.5.
(i) If filters F, F ′ satisfy F (xi) ≤ F
′(xi) for all i = 1, . . . , n, then c(a, F ) ≤
c(a, F ′) for all non-zero classes a ∈ H(V).
(ii) If F is a filter and θ ∈ R, then F + θ is again a filter and c(a, F + θ) =
c(a, F ) + θ for all non-zero classes a ∈ H(V).
The proof is obvious and we omit it. It follows that for any two filters F, F ′
|c(a, F )− c(a, F ′)| ≤ ||F − F ′||C0 ∀a ∈ H(V) \ {0} .
Assume now that V1,V2 are decorated complexes. Denote by F1, F2 their
filters. Fix ǫ > 0. By a small perturbation of the filters we get new filters,
F ′1 and F
2, on our complexes so that the complexes become generic and in
general position, and furthermore
||F1 − F
1||C0 ≤ ǫ , ||F2 − F
2||C0 ≤ ǫ .
Given homology classes ai ∈ H(Vi) we have
|c(a1, F1) + c(a2, F2)− c(a1 ⊗ a2, F1 + F2)| ≤
|c(a1, F
1) + c(a2, F
2)− c(a1 ⊗ a2, F
1 + F
2)|+ 4ǫ = 4ǫ .
Here we used that Theorem 5.2 is already proved for generic complexes in
general position. Since ǫ > 0 is arbitrary, we get that
c(a1, F1) + c(a2, F2)− c(a1 ⊗ a2, F1 + F2) = 0 ,
which completes the proof of Theorem 5.2 in full generality.
6 Stable non-displaceability of heavy sets
In this section we prove part (ii) of Theorem 1.2.
Proposition 6.1. Every heavy subset is stably non-displaceable.
For the proof we shall need the following auxiliary statement. Given R > 0,
consider the torus T2R obtained as the quotient of the cylinder T
∗S1 = R(r)×
S1 (θ mod 1) by the shift (r, θ) 7→ (r + R, θ). For α > 0 define the function
Fα(r, θ) := αf(r) on T
R, where f(r) is any R-periodic function having only
two non-degenerate critical points on [0, R]: a maximum point at r = 0 with
f(0) = 1, and a minimum point at r = R/2, f(R/2) =: −β < 0. We denote
by [T ] the fundamental class of T2R. We work with the symplectic form dr∧dθ
on T2R.
Lemma 6.2. c([T ], Fα) = α.
Proof. Note that the contractible closed orbits of period 1 of the Hamiltonian
flow generated by Fα are fixed points forming circles S+ = {r = 0} and
S− = {r = R/2}. The actions of the fixed points on S± equal respectively to
α and −αβ, and thus the spectral invariants of Fα lie in the set {α,−αβ}.
Recall from [59] that c([T ], Fα) > c([point], Fα). Thus c([T ], Fα) = α.
Lemma 6.3. Let H ∈ C∞(M) so that H−1(maxH) is displaceable. Then
ζ(H) < maxH.
Proof. Choose ǫ > 0 so that the set
H−1((maxH − ǫ,maxH ])
is displaceable. Choose a real-valued cut-off function ρ : R → [0, 1] which
equals 1 near maxH and which is supported in (maxH−ǫ,maxH+ǫ). Thus
ρ(H) is supported in H−1((maxH − ǫ; maxH ]) and ζ(ρ(H)) = 0. Since H
and ρ(H) Poisson-commute, the vanishing and the monotonicity axioms yield
ζ(H) = ζ(ρ(H)) + ζ(H − ρ(H)) ≤ max(H − ρ(H)) < maxH .
Proof of Proposition 6.1: It suffices to show that for every R > 0 the set
Y := X × {r = 0} ⊂M ′ :=M × T2R
is non-displaceable. Assume on the contrary that Y is displaceable. Choose
a function H on M with H ≤ 0, H−1(0) = X . Put
H ′ = H + F1 = H + f(r) :M
′ → R.
Assume that the partial quasi-state ζ on M is associated to some non-zero
idempotent a ∈ QH∗(M) by means of (2). Denote by ζ
′ the quasi-state on
M ′ associated to a⊗ T . Note that
Y = (H ′)−1(maxH ′) , where maxH ′ = 1 ,
while Theorem 5.1 and Lemma 6.2 imply that
ζ ′(H ′) = ζ(H) + 1 .
By Lemma 6.3 ζ ′(H ′) < 1 and so ζ(H) < 0. In view of Proposition 4.1, we
get a contradiction with the heaviness of X .
Lemma 6.2 also yields a simple proof of the following result which also follows
from Corollary 1.15:
Corollary 6.4. Any meridian of T2 is heavy (with respect to the fundamental
class [T ]).
Proof. In the notation as above identify T2 with T21 for R = 1. Since any
two meridians of T2 can be mapped into each other by a symplectic isotopy
and since such an isotopy preserves heaviness, it suffices to prove that the
meridian S := S+ = {r = 0} (see the proof of Lemma 6.2) is heavy.
Let H : T2 → R be a Hamiltonian and let us show that ζ(H) ≥ infSH ,
where ζ is defined using [T ]. Shifting H , if necessary, by a constant, we may
assume without loss of generality that infSH = 1. Pick f = f(r) : T
2 → R
as in the definition of Fα so that F1 = f ≤ H on T
2 (note that f equals 1 on
S). Then Lemma 6.2 yields
ζ(H) ≥ ζ(F1) = 1 = inf
7 Analyzing stable stems
Proof of Theorem 1.6: Assume that A is a Poisson-commutative subspace
of C∞(M), Φ : M → A∗ its moment map with the image ∆, and let X =
Φ−1(p) be a stable stem of A.
Take any functionH ∈ C∞(A∗) with H ≥ 0 andH(p) = 0. We claim that
ζ(Φ∗H) = 0. By an arbitrarily small C0-perturbation of H we can assume
that H = 0 in a small neighborhood, say U , of p. Choose an open covering
U0, U1, . . . , UN of ∆ so that U0 = U , and all Φ
−1(Ui) are stably displaceable
for i ≥ 1 (it exists by the definition of a stem). Let ρi : ∆ → R, i = 0, . . . , N ,
be a partition of unity subordinated to the covering {Ui}.
Take the two-torus T2R as in Section 6. Choose R > 0 large enough so that
Φ−1(Ui)× {r = const} is displaceable in M × T
R for all i ≥ 1. Choose now
a sufficiently fine covering Vj , j = 1, . . . , K, of the torus T
R by sufficiently
thin annuli {|r − rj | < δ} so that the sets Φ
−1(Ui) × Vj are displaceable in
M × T2R for all i ≥ 1 and all j. Let ̺j = ̺j(r), j = 1, . . . , K, be a partition
of unity subordinated to the covering {Vj}.
Denote by ζ ′ the partial quasi-state corresponding to a⊗T . Put F (r, θ) =
cos(2πr/R). Write
Φ∗H + F =
(Φ∗H + F ) · Φ∗ρi · ̺j =
Φ∗(Hρ0) + F · Φ
∗ρ0 +
(Φ∗H + F ) · Φ∗ρi · ̺j .
Note that Hρ0 = 0 and F · Φ
∗ρ0 ≤ 1. Applying partial quasi-additivity
and monotonicity we get that
ζ ′(Φ∗H + F ) = ζ ′(F · Φ∗ρ0) ≤ 1.
By Lemma 6.2 and the product formula (Theorem 5.1 above) we have
ζ ′(Φ∗H + F ) = ζ(Φ∗H) + 1 ≤ 1
and hence ζ(Φ∗H) ≤ 0. On the other hand, ζ(Φ∗H) ≥ 0 since H ≥ 0. Thus
ζ(Φ∗H) = 0 and the claim follows.
Further, given any function G on M with G ≥ 0 and G|X = 0, one can
find a function H on A∗ with H(p) = 0 so that G ≤ Φ∗H . By monotonicity
and the claim above
0 ≤ ζ(G) ≤ ζ(Φ∗H) = 0 ,
and hence ζ(G) = 0. Thus X is superheavy.
8 Monotone Lagrangian submanifolds
The main tool of proving (super)heaviness of monotone Lagrangian subman-
ifolds satisfying the Albers condition is the spectral estimate in Proposi-
tion 8.1(iii) below, which originated in the work by Albers [2]. Later on
Biran and Cornea pointed out a mistake in [2], and suggested a correction
together with a far reaching generalization in [15]. Let us mention that the
original Albers estimate was used in the first version of the present paper. We
thank Biran and Cornea for informing us about the mistake, explaining to
us their approach and helping us to correct a number of our results affected
by this mistake.
The main ingredient of Biran-Cornea techniques which is needed for our
purposes is the following result. Let (M,ω) be a closed monotone symplectic
manifolds with [ω] = κ·c1(M), κ > 0. WriteN for the minimal Chern number
of (M,ω). Let Ln ⊂M2n be a closed monotone Lagrangian submanifold with
the minimal Maslov number NL ≥ 2.
We shall treat slightly differently the cases when NL is even and odd. Let
us mention that for orientable L, NL is automatically even. Thus, due to
our convention, when NL is odd we work with the basic field F = Z2. Let
Γ = κN · Z be the group of periods of M . Recall that the quantum ring has
the form QH∗(M) = H∗(M ;F) ⊗F Λ, where the Novikov ring Λ is defined
as Λ = KΓ[q, q
−1] . Put Γ′ = (κN/2) · Z. Consider an extended Novikov ring
Λ′ := KΓ′ [q
2 , q−
2 ]. Define now QH ′∗(M) as QH∗(M) if NL is even, and as
H∗(M,Z2)⊗Z2 Λ
′ if NL is odd. In the latter case QH
∗(M) is an extension of
QH∗(M), and we shall consider without special mentioning QH∗(M), Λ, KΓ
as subrings of QH ′∗(M), Λ
′, KΓ′. The grading of QH
∗(M) is determined by
the condition deg q
2 = 1. As before, we shall use notation QH ′•(M), where
• = “even” when F = C and • = ∗ when F = Z2.
Note that the spectral invariants (and hence partial symplectic quasi-
states) are well-defined over the extended ring, and furthermore, their values
and properties, by tautological reasons, do not alter under such an extension
(cf. a discussion in [15], Section 5.4). Put w := sκNL/2qNL/2. Recall that j
stands for the natural morphism H•(L;F) → H•(M ;F).
Proposition 8.1. Assume that k > n+1−NL. If F = C assume in addition
that k is even. Then there exists a canonical homomorphism jq : Hk(L;F) →
QH ′k(M) with the following properties
8The letter “q” in jq stands for quantum.
(i) jq(x) = j(x) + w−1y, where y is a polynomial in w−1 with coefficients
in H•(M ;F);
(ii) jq([L]) = j([L]);
(iii) If jq(x) 6= 0 then c(jq(x), H) ≤ supLH for every H ∈ C
∞(M).
In particular, if S is an Albers element of L, we have jq(S) = j(S)+O(w−1) 6=
This proposition was proved by Biran and Cornea in [15] in the case
F = Z2: The map j
q is essentially the map iL appearing in Theorem A(iii)
in [15]. Proposition 8.1(i) above is a combination of Theorem A(iii) and
Proposition 4.5.1(i) in [15]. Our variable w corresponds to the variable t−1 in
[15], while our sNκqN corresponds to the variable s−1 in Section 2.1.2 of [15].
After such an adjustment of the notation, the formula w := sκNL/2qNL/2 above
can be extracted from Section 2.1.2 of [15]. For Proposition 8.1(ii) above
see Remark 5.3.2.a in [15]. Proposition 8.1(iii) above follows from Lemma
5.3.1(ii) in [15]. Finally, let us repeat the disclaimer made in Section 1.5: we
take for granted that Proposition 8.1 remains valid for the case F = C.
Let us pass to the proofs of our results on (super)-heaviness of monotone
Lagrangian submanifolds. We start with the following remark. Let S be an
Albers element of L. The Poincaré duality property of spectral invariants
(see Section 3.4 above) extends verbatim to the case of the ring QH ′(M)
with the following modification: When NL is odd, the pairing Π introduced
in Section 3.4 extends in the obvious way to a non-degenerate F -valued
pairing on QH ′•(M) which we still denote by Π. Applying Poincaré duality
and substituting H := −F into Proposition 8.1 (iii) above we get that for
every function F ∈ C∞(M)
c(T, F ) ≥ inf
F ∀T ∈ QH ′•(M) with Π(T, j
q(S)) 6= 0.
In particular, given a non-zero idempotent a ∈ QH ′•(M) and a class b ∈
QH ′•(M), so that Π(a∗b, j
q(S)) 6= 0, we get, using the normalization property
of spectral invariants, that
c(a, F ) + ν(b) ≥ c(a ∗ b, F ) ≥ inf
F ∀F ∈ C∞(M) . (27)
Therefore, applying (27) to kF for k ∈ N, dividing by k and passing to the
limit as k → +∞, we get that for the partial quasi-state ζ , defined by a,
ζ(F ) ≥ inf
F ∀F ∈ C∞(M),
meaning that L is heavy with respect to a.
Proof of Theorem 1.15: Let S be an Albers element of L. Let T ∈
H•(M ;F) be any singular homology class such that T ◦ j(S) 6= 0. Thus,
applying Proposition 8.1 (i) we see that Π([M ]∗T, jq(S)) = Π(T, jq(S)) 6= 0,
and hence inequality (27), applied to a = [M ], b = T , yields that L is heavy
with respect to [M ].
Let us pass to the proof of Theorem 1.25 on the effect of semi-simplicity
of the quantum homology. It readily follows from the next more general
statement. Let L1, . . . , Lm be Lagrangian submanifolds satisfying the Albers
condition. Let Si be any Albers element of Li. Denote by Zi = j
q(Si) ∈
QH ′•(M) its image under the inclusion morphism from Proposition 8.1 above.
Theorem 8.2. Given such L1, . . . , Lm and Z1, . . . , Zm, assume, in addition,
that QH2n(M) is semi-simple and the Lagrangian submanifolds L1, . . . , Lm
are pair-wise disjoint. Then the classes Z1, . . . , Zm are linearly independent
over KΓ′.
Proof. Arguing by contradiction, assume that
Z1 = α2Z2 + . . .+ αmZm (28)
for some α2, . . . , αm ∈ KΓ′ . Since QH2n(M) is semi-simple, it decomposes
into a direct sum of fields with unities e1, . . . , ed. Since the pairing Π (on
QH ′•(M ;F)) is non-degenerate, there exists T ∈ QH
•(M ;F) such that
Π(T, Z1) 6= 0. (29)
Let us write T as
T = [M ] ∗ T =
ei ∗ T. (30)
Equations (29), (30) imply that there exists l ∈ [1, d] such that
Π(el ∗ T, Z1) 6= 0 . (31)
Then (28) implies that there exists r ∈ [2, m] such that
Π(el ∗ T, αrZr) 6= 0.
Using (21) (for Π on QH ′•(M ;F)) we can rewrite the last equation as
Π(el ∗ αrT, Zr) 6= 0. (32)
Applying now formula (27) for S = Z1 ∈ H•(L1;F), a = el, b = T , and also
for S = Zr ∈ H•(Lr;F), a = el, b = αrT , we conclude that both L1 and Lr
are heavy with respect to el. Thus they are superheavy with respect to el,
because el is the unity in a field factor of QH2n(M) (see Section 1.6). Hence
they must intersect – in contradiction to the assumption of the theorem. This
finishes the proof of the first part of the theorem.
Proof of Theorem 1.25(a): Assume that L1, . . . , Lm are pair-wise disjoint
Lagrangian submanifolds satisfying the condition (a) from the formulation
of the theorem. Denote by Ni the minimal Maslov number of Li. Since
Ni > n + 1, the class of a point from H0(Li;F) is an Albers element for Li.
Let Zi ∈ QH
0(M) be its image under the Biran-Cornea inclusion morphism
associated to Li. Note that by Proposition 8.1(i) Zi = p + aiw
i , where
wi = s
κNi/2qNi/2, ai ∈ HNi(M ;F) and p ∈ H0(M ;F) is the homology class of
a point. Observe that degwi = Ni > n + 1, and hence the expression for Zi
cannot contain terms in w−1i of order two and higher, since HkNi(M ;F) = 0
for k ≥ 2.
Recall now that all Ni’s lie in some set Y of positive integers. Let W ⊂
QH ′0(M) be the span over KΓ′ of
H0(M ;F)⊕
s−κE/2q−E/2 ·HE(M ;F) .
Note that
dimKΓ′ W = βY (M) + 1 < m .
Thus the elements Zi, i = 1, . . . , m, are linearly dependent over KΓ′ . By
Theorem 8.2, QH2n(M) is not semi-simple.
Proof of Theorem 1.25(b): Assume that L1, . . . , Lm are pair-wise disjoint
homologically non-trivial Lagrangian submanifolds. By Proposition 8.1(ii)
jq([Li]) = j([Li]) for every i = 1, . . . , m. Since the classes j([Li]) are linearly
dependent, Theorem 8.2 implies that QH2n(M) is not semi-simple.
Proof of Theorem 1.18: Combining Proposition 8.1 (ii) and (iii) we get
that for any H ∈ C∞(M)
c(j([L]), H) ≤ sup
H ∀H ∈ C∞(M) .
By the hypothesis of the theorem, we can write j([L]) ∗ b = a for some b.
c(a,H) = c(j([L]) ∗ b,H) ≤ c(j([L]), H) + c(b, 0) .
c(a,H) ≤ sup
H + c(b, 0) .
Applying this inequality to E · H with E > 0, dividing by E and passing
to the limit as E → +∞ we get that ζ(H) ≤ supLH for all H . Thus L is
superheavy.
Remark 8.3. The results above admit the following generalizations in the
framework of the Biran-Cornea theory. The main object of this theory is the
quantum homology ring QH∗(L) of a monotone Lagrangian submanifold,
which is isomorphic to the Lagrangian Floer homology HF∗(L, L) of L up to
a shift of the grading.
(i) If QH∗(L) does not vanish then L is heavy (see Remark 1.2.9b in [15]).
In fact, it follows from [15] that if L satisfies the Albers condition,
QH∗(L) does not vanish.
(ii) The map jq of the Proposition 8.1 above is a footprint of the quan-
tum inclusion map iL : QH∗(L) → QH
∗(M) constructed in [15]. The
analogue of the action estimate in item (iii) of the proposition is ob-
tained in [15] for the classes iL(x) for elements x ∈ QH∗(L) of a certain
special form, yielding the following generalization of Theorem 1.18: for
these special classes x ∈ QH∗(L) the condition that the class iL(x)
does not vanish and divides a non-trivial idempotent a implies that L
is superheavy with respect to a. This enables, for instance, to general-
ize Example 1.19 on Lagrangian spheres in quadrics above to the case
when dimL is odd.
(iii) In [15] one can find another action estimate which comes from the
QH∗(M)-module structure on QH∗(L), which yields more results on
(super)heaviness of monotone Lagrangian submanifolds.
Proof of Proposition 1.4: The quantum homology QH2n(M) splits as an
algebra over K into a direct sum of two algebras one of which is a field. This
was proved by McDuff for the field F = C (see [39] and [24, Section 7]), but
the proof goes through for the case F = Z2 as well. Denote the unity of the
field by a. It is a non-zero idempotent in QH2n(M). As we already pointed
out in Remark 1.21, such an idempotent a defines a genuine symplectic quasi-
state and therefore the classes of heavy and superheavy sets with respect to
a coincide.
By Theorem 1.2, the Lagrangian torus L ⊂ M cannot be superheavy
with respect to a, since it can be displaced from itself by a symplectic (non-
Hamiltonian) isotopy. Indeed, take an obvious symplectic isotopy φt of T
that displaces L (a parallel shift) and compose it with a Hamiltonian isotopy
ψt so that for every t we have that ψt is constant on φt(L) and ψtφt is identity
on the ball where the blow up of T2n was performed. Clearly, the resulting
symplectic isotopy ψtφt extends to a symplectic isotopy of M that displaces
On the other hand, NL ≥ 2 because in this case NL = 2N , where N ≥ 1
is the minimal Chern number of M . Finally, note that L represents a non-
trivial homology class in Hn(M ;Z2). Therefore we can apply Theorem 1.15
and get that L is heavy with respect to [M ].
9 Rigidity of special fibers of Hamiltonian ac-
tions
In this section we prove Theorem 1.9. Denote the special fiber of Φ by
L := Φ−1(pspec).
Reduction to the case of T1-actions: First, we claim that it is enough
to prove the theorem for Hamiltonian T1-actions and the general case will
follow from it. Indeed, assume this is proved. The superheaviness of the
special fiber immediately yields that for any function H̄ : R → R
ζ(Φ∗H̄) = H̄(pspec), (33)
where Φ :M → R is the moment map of the T1-action.
Let us turn to the multi-dimensional situation and let Φ : M → Rk
be the normalized moment map of a Hamiltonian Tk-action on M . For a
v ∈ Rk denote by Φv(x) = 〈v,Φ(x)〉, where 〈·, ·〉 is the standard Euclidean
inner product on Rk. Note that if v ∈ Zk the function Φv is the normalized
moment map of a Hamiltonian circle action and its special value is 〈v, pspec〉.
Thus by (33)
K) = K(〈v, pspec〉) ∀K ∈ C
∞(R) . (34)
By homogeneity of ζ , equality (34) holds for all v ∈ Qk, and by continuity
for all v ∈ Rk.
Observe that for each pair of smooth functions P,Q ∈ C∞(R) and for each
pair of vectors v,w ∈ Rk the functions Φ∗
P and Φ∗
Q Poisson-commute on
M . Thus the triangle inequality for the spectral numbers (see Section 3.4)
yields
P + Φ∗
Q) ≤ ζ(Φ∗
P ) + ζ(Φ∗
Q) . (35)
Since M is compact, it suffices to assume that the function H̄ ∈ C∞(Rk) on
Rk is compactly supported. By the inverse Fourier transform we can write
H̄(p) =
sin〈v, p〉 · F (v) + cos〈v, p〉 ·G(v)
for some rapidly (say, faster than (|p| + 1)−N for any N ∈ N) decaying
functions F and G on Rk. For every v ∈ Rk define a function Kv ∈ C
Kv(s) := sin s · F (v) + cos s ·G(v) .
Observe that
Φ∗H̄ =
Kv dv .
Denote by B(R) the Euclidean ball of radius R in Rk with the center at the
origin. Put
H̄R(p) =
Kv(〈v, p〉) dv, p ∈ R
Since the functions F and G are rapidly decaying, we get that
||H̄R − H̄||C0(Rk) → 0 as R → ∞ . (36)
We claim that for every R
ζ(Φ∗H̄R) ≤ H̄R(pspec) . (37)
Indeed, for ǫ > 0 introduce the integral sum
H̄R,ε(p) =
v∈ ε·Zk∩B(R)
εk ·Kv(〈v, p〉) .
Φ∗H̄R,ε =
v∈ ε·Zk∩B(R)
εk · Φ∗
Applying repeatedly (35) and (34) we get that
ζ(Φ∗H̄R,ε) ≤ H̄R,ε(pspec) .
Note now that for fixed R the family H̄R,ǫ converges to H̄R as ε → 0 in
the uniform norm on C0(Rk). Using that ζ is Lipschitz with respect to the
uniform norm on C0(M) we readily get the inequality (37).
Combining the fact that ζ is Lipschitz with (36) and (37) we get that
ζ(Φ∗H̄) = lim
ζ(Φ∗H̄R) ≤ lim
H̄R(pspec) = H̄(pspec) .
Now, assume that H̄ ≥ 0 and H̄(pspec) = 0. We just have proved that
ζ(Φ∗H̄) ≤ 0, and hence ζ(H) = 0, which immediately yields the desired su-
perheaviness of the special fiber. This completes the reduction of the general
case to the 1-dimensional case.
From now on we will consider only the case of an effective Hamil-
tonian T1-action on M with a moment map Φ :M → R. Its moment
polytope ∆ is a closed interval in R and pspec = −I(Φ) ∈ R.
Reduction to the case of a strictly convex function: We claim
that it is enough to show the following proposition:
Proposition 9.1. Assume H̄ : R → R is a strictly convex smooth function
reaching its minimum at pspec. Set H := Φ
∗H̄. Then ζ(H) = H̄(pspec).
Postponing the proof of the proposition for a moment let us show that it
implies the theorem. Indeed, let F : M → R be a Hamiltonian on M . In
order to show the superheaviness of L = Φ−1(pspec) we need to show that
ζ(F ) ≤ supL F . Pick a very steep strictly convex function H̄ : R → R with
the minimum value supL F reached at pspec and such that Φ
∗H̄ =: H ≥ F
everywhere on M . Then using Proposition 9.1 and the monotonicity of ζ we
ζ(F ) ≤ ζ(H) = H̄(pspec) = sup
yielding the claim.
Preparations for the proof of Proposition 9.1: Given a (time-
dependent, not necessarily regular) Hamiltonian G, we associate to every
pair [γ, u] ∈ P̃G a number
DG([γ, u]) := AG([γ, u])−
· CZG([γ, u]).
(Recall that we defined the Conley-Zehnder index for all Hamiltonians and
not only the regular ones – see Section 3.3). The number DG([γ, u]) is in-
variant under a change of the spanning disc u – an addition of a sphere
jS ∈ HS2 (M) to the disc u changes both AG([γ, u]) and κ/2 · CZG([γ, u]) by
the same number. Thus we can write DG([γ, u]) = DG(γ).
Given [γ, u] ∈ P̃G and l ∈ N define γ
(l) and u(l) as the compositions
of γ and u with the map z → zl on the unit disc D2 ⊂ C (here z is a
complex coordinate on C). Denote by t 7→ gt the time-t flow of G and by
G(l) :M × R → R the Hamiltonian whose time-t flow is t 7→ (gt)
l and which
is defined by
G(l) := G♯ . . . ♯G (l times),
where G♯K(x, t) := G(x, t) +K(g−1t x, t) for any K :M × R → R.
Proposition 9.2. There exists a constant C > 0, depending only on n, with
the following property. Given a 1-periodic orbit γ ∈ PG of the flow t 7→ gt
generated by G, assume that γ(l) is a 1-periodic orbit of the flow t 7→ glt
generated by G(l), and therefore for any u such that [γ, u] ∈ P̃G we have
[γ(l), u(l)] ∈ P̃G(l). Then
|DG(l)([γ
(l), u(l)])− lDG([γ, u])| ≤ l · C.
Proof. The action term in DG gets multiplied by l as we pass from G to G
As for the Conley-Zehnder term, the quasi-morphism property of the Conley-
Zehnder index (see Proposition 3.5) implies that there exists a constant C > 0
(depending only on n) such that
|lCZG[γ, u]− CZG(l)([γ
(l), u(l)])| ≤ C.
This immediately proves the proposition.
Proposition 9.3. Let G :M × [0, 1] → R be a Hamiltonian as above. Then
one can choose ǫ > 0, depending on G, and a constant Cn > 0, depending
only on n = dimM/2, so that any function F : M × [0, 1] → R which is ǫ-
close to G in a C∞-metric on C∞(M×[0, 1]) satisfies the following condition:
for every γ0 ∈ PF there exists γ ∈ PG such that the difference between DF (γ0)
and DG(γ) is bounded by Cn.
Proof. Denote the flow of G by gt (as before) and the flow of F by ft. We
will view time-1 periodic trajectories of these flows both as maps of [0, 1] to
M having the same value at 0 and 1 and as maps from S1 to M .
First, consider the fibration D2×M →M and, slightly abusing notation,
denote the natural pullback of ω again by ω. Second, look at the fibration
pr : D2 ×M → D2. Denote by V ert the vertical bundle over D2 ×M formed
by the tangent spaces to the fibers of pr. For each loop σ : S1 →M define by
σ̂ : S1 → D2 ×M the map σ̂(t) := (t, γ(t)). The bundles σ∗TM and σ̂∗V ert
over S1 coincide. Similarly for each w : D2 →M denote by ŵ : D2 → D2×M
the map ŵ(z) := (z, w(z)).
There exists δ > 0, depending on G, such that for each γ ∈ PG a tubular
δ-neighborhood of the image of γ̂ in S1 ×M ⊂ D2 ×M , denoted by Ubγ, has
the following properties:
• there exists a 1-form λ on Ubγ satisfying dλ = ω;
• V ert admits a trivialization over Ubγ .
Given an ǫ > 0, we can choose F sufficiently C∞-close to G so that the
paths t 7→ ft and t 7→ gt in Ham(M) are arbitrarily C
∞-close and therefore
• for every x ∈ Fix (F ) there exists y ∈ Fix (G) which is ǫ-close to x
(think of the fixed points as points of intersection of the graph of a
diffeomorphism with the diagonal);
• the C∞-distance between the maps γ0 : t 7→ ft(x) and γ : t 7→ gt(y)
from [0, 1] to M is bounded by ǫ and the image of γ̂0 lies in Ubγ.
Pick a map u0 : D
2 → M , u|∂D2 = γ0. Since γ0 and γ are C
∞-close one
can enlarge D2 to a bigger disc D21 ⊃ D
2 and find a smooth map u : D21 →M
so that
• u|∂D21
= u0;
• u(D21 \ D
2) ⊂ Ubγ.
Rescaling D21 we may assume without loss of generality that [γ, u] ∈ PG.
Trivialize the vector bundles γ∗0TM and γ
∗TM so that the trivializations
extend to a trivialization of u∗TM over D21 (and hence of u
0TM over D
Using the trivializations we can identify the paths t 7→ dγ0(0)ft and t 7→ dγ(0)gt
with some identity-based paths of symplectic matrices A(t), B(t). Fixing a
small ǫ as above, we can also assume that F is chosen so C∞-close to G that,
in addition to all of the above, the C∞-distance between the paths t 7→ A(t)
and t 7→ B(t) in Sp (2n) is bounded by ǫ (for instance, make sure first that
the matrix paths obtained by writing the paths t 7→ dγ0(0)ft and t 7→ dγ(0)gt
using some trivialization of V ert over Ubγ are close enough – then the matrix
paths t 7→ A(t) and t 7→ B(t) will also be close enough).
We claim that by choosing ǫ sufficiently small in the construction above we
can bound the difference between DF ([γ0, u0]) and DG([γ, u]) by a quantity
depending only on dimM .
Indeed, the difference |
F (γ0(t), t)dt −
G(γ(t))dt| is bounded by a
quantity depending only on some universal constants and ǫ, because γ0 is
ǫ-close to γ and F is ǫ-close to G with respect to the C∞-metrics. It can be
made arbitrarily small by choosing a sufficiently small ǫ. The difference
u∗0ω −
u∗ω| = |
û∗0ω −
û∗ω|
is bounded by the difference |
γ̂∗0λ −
γ̂∗λ|. Since, γ0 and γ are ǫ-close
in the C∞-metric the later difference can be made less than 1 if we choose
a sufficiently small ǫ. Thus we have shown that by choosing a sufficiently
small ǫ we can bound |AF ([γ0, u0])−AG([γ, u])| by 1.
Now, as far as the Conley-Zehnder indices are concerned, our choice
of the trivializations means that the difference between CZF ([γ0, u0]) and
CZG([γ, u]) is just the difference between the Conley-Zehnder indices for the
matrix paths t 7→ A(t) and t 7→ B(t). But the latter paths in Sp (2n) are
ǫ-close in the C∞-sense, hence represent close elements of S̃p (2n) and if ǫ
was chosen sufficiently small, then, as we mentioned in Section 3.3, their
Conley-Zehnder indices differ at most by a constant depending only on n.
This finishes the proof of the claim and the proposition.
Plan of the proof of Proposition 9.1: We assume now that H̄ is
a fixed strictly convex function on R. Our calculations will feature E as a
large parameter. For quantities α, β depending on E we will write α � β if
α ≤ β+const holds for large enough E, where const depends only on (M,ω),
Φ and H̄ , and in particular does not depend on E. We will write α ≈ β if
α � β and β � α. Using this language the proposition can be restated as
c(a, EH) ≈ EH̄(pspec). (38)
In general, 1-periodic orbits of the flow of EH are not isolated and there-
fore the Hamiltonian is not regular. Let F be a regular (time-periodic)
perturbation of EH .
By the spectrality axiom, the spectral number c(a, F ) for a ∈ QH2n(M)
equals AF ([γ0, u0]) for some pair [γ0, u0] ∈ P̃F with CZF ([γ0, u0]) = 2n. Thus
c(a, F ) ≈ DF (γ0). Combining this with Proposition 9.3 we get that for some
γ ∈ PEH
EH̄(pspec) � c(a, EH) ≈ c(a, F ) ≈ DF (γ0) ≈ DEH(γ) . (39)
Thus it would be enough to show that
DEH(γ) � EH̄(pspec) for all γ ∈ PEH , (40)
which together with (39) would imply (38).
Inequality (40) will be proved in the following way. Note that each γ ∈
PEH lies in Φ
−1(p) for some p ∈ ∆. We will show that
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p). (41)
Note that (41) implies (40). Indeed, since H̄ is strictly convex and reaches
its minimum at pspec, it follows from (41) that
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p) ≤ EH̄(pspec),
which is true for any γ ∈ PEH thus yielding (40).
Proof of (41): Let the T1-action on M be given by a loop of sym-
plectomorphisms {φt}, t ∈ R, φt = φt+1. The flow of EH has the form
htx = φEH̄′(Φ(x))tx.
We view γ as a map γ : [0, 1] → M satisfying γ(0) = γ(1). Denote
x := γ(0). The curve γ lies in Φ−1(p).
Denote N := γ([0, 1]). This is the T1-orbit of x and it is either a point or
a circle.
In the first case γ is a constant trajectory concentrated at a fixed point
N ∈M of the action. Using this constant curve γ together with the constant
disc u spanning for the definitions of I(Φ) and DEH(γ) one gets
pspec − p = mΦ(γ, u) · κ/2,
DEH(γ) = EH̄(p)− κ/2 · CZEH([γ, u]).
Thus proving (41) reduces in this case to proving
−CZEH([γ, u]) ≈ EH̄
′(p) ·mΦ(γ, u).
Let us fix a symplectic basis of TNM and view each differential dNφt as a
symplectic matrix A(t), so that {A(t)} is an identity-based loop in Sp (2n).
−CZEH([γ, u]) ≈ CZmatr({A(EH̄
′(p)t)}),
while
EH̄ ′(p) ·mΦ(γ, u) ≈ EH̄
′(p)Maslov({A(t)}).
Thus we need to prove
CZmatr({A(EH̄
′(p)t)}) ≈ EH̄ ′(p)Maslov({A(t)}),
which follows easily from the definitions of the Conley-Zehnder index and
the Maslov class.
Thus from now on we will assume that N is a circle. Take any point
x ∈ N . The stabilizer of x under the T1-action is a finite cyclic group of
order k ∈ N. Thus the orbit of the T1-action turns k times along N . Since γ
is a non-constant closed orbit of the Hamiltonian flow generated by EΦ∗H̄ ,
it turns r times along N with r ∈ Z \ {0}. This implies that EH̄ ′(p) = r/k.
We claim that without loss of generality we may assume that l := r/k is an
integer.
Indeed, we can always pass to γ(k) ∈ PkEH , so that (kEH̄)
′(p) ∈ Z, and
if we can prove the proposition for γ(k), then
DkEH(γ
(k)) ≈ kEH̄(p) + kEH̄ ′(p)(pspec − p).
Applying Proposition 9.2 we get
kDEH(γ) ≈ kEH̄(p) + kEH̄
′(p)(pspec − p) + k · const ,
and hence
DEH(γ) ≈ EH̄(p) + EH̄
′(p)(pspec − p),
proving the claim for the original γ.
From now on we assume that l := EH̄ ′(p) ∈ Z\{0} and that [γ, u] ∈ P̃lΦ.
Consider the Hamiltonian vector field X := sgradΦ at a point x ∈ N . Since
N is a non-constant orbit we get X 6= 0. Then V = Tx(Φ
−1(p)) is the skew-
orthogonal complement to X . Choose a T1-invariant ω-compatible almost
complex structure J in a neighborhood of N . Together ω and J define a T1-
invariant Riemannian metric g. Decompose the tangent bundle TM along
N as follows. Put Z = Span(JX,X) and set W to be the g-orthogonal
complement to X in V . Thus we have a T1-invariant decomposition
TxM = W ⊕ Z , x ∈ N . (42)
Furthermore, W and Z carry canonical symplectic forms. Thus W and Z
define symplectic (and hence trivial) subbundles of TM over N . They induce
trivial subbundles of the bundle γ∗TM over S1.
We calculate
dht(x)ξ = dφEH′(Φ(x))t(x)ξ + EH
′′(Φ(x)) · dΦ(ξ) ·X . (43)
We consider two trivializations of the bundle γ∗TM over S1. The first trivi-
alization is defined by means of sections invariant under the T1-action. The
second one is chosen in such a way that it extends to a trivialization of u∗TM
over D2. Using these trivializations we can identify dht(x), respectively, with
two identity-based paths {Ct}, {C
t} of symplectic matrices. The decompo-
sition (42) induces a split
Ct = 1⊕ Bt .
We claim that |CZmatr({Bt})| is bounded by a constant independent of E.
Indeed, observe that in the basis (X, JX) of Z
1 b12(t)
Denote by L the line spanned by X = (1, 0). Perturb {Bt} to a path {B
RδtBt}, where Rt is the rotation by angle t, and δ > 0 is small enough.
Observe thatB′(t)L∩L = {0} for t > 0. It follows readily from the definitions
that |CZmatr(B
t)| and |CZmatr(Rδt)| do not exceed 2. Thus by the quasi-
morphism property of the Conley-Zehnder index (see Proposition 3.5) we
have that |CZmatr({Bt})| is bounded by a constant independent of E, which
yields the claim. Therefore
CZmatr ({Ct}) ≈ 0 .
On the other hand, by formula (18)
CZmatr ({C
t}) = CZmatr ({Ct}) +mlΦ([γ, u]) .
CZEH([γ, u]) := n− CZmatr ({C
t}) ≈ −mlΦ([γ, u]). (44)
Since the periodic trajectory γ lies inside Φ−1(p), we get
AEH([γ, u]) =
EH(γ(t))dt−
u∗ω = EH̄(p)−
u∗ω. (45)
Using (45) and (44) the precise equality
DEH([γ, u]) = AEH([γ, u])−
· CZEH([γ, u])
can be turned into an asymptotic inequality
DEH([γ, u]) ≈ EH̄(p)−
u∗ω +
mlΦ([γ, u]). (46)
Since the periodic trajectory γ lies inside Φ−1(p), we have
AlΦ([γ, u]) =
lΦ(γ(t))dt−
u∗ω = lp−
u∗ω. (47)
Adding and subtracting lp from the right-hand side of (46) and using (47)
we get
DEH(γ) = DEH([γ, u]) ≈
EH̄(p)− lp)
mlΦ([γ, u])
EH̄(p)− lp
AlΦ([γ, u])+
mlΦ([γ, u])
EH̄(p)− lp
− I(lΦ) =
= EH̄(p) + l(−I(Φ)− p) = EH̄(p) + l(pspec − p) .
Recalling that l = EH ′(p), we finally obtain that
DEH(γ) = EH̄(p) + EH
′(p)(pspec − p),
which is precisely the equation (41) that we wanted to get. This finishes the
proof of Proposition 9.1 and Theorem 1.9.
9.1 Calabi and mixed action-Maslov
Proof of Theorem 1.13.
Assume H :M × [0, 1] → R is a normalized Hamiltonian which generates
a loop in Ham(M) representing a class α ∈ π1(Ham(M)) ⊂ H̃am (M). Then
H(l) is also normalized and generates a loop representing αl. Let us compute
µ(α) = −vol (M) · liml→+∞ c(a,H
(l))/l.
Arguing as in the proof of (39) we get that there exists a constant C >
0 such that for each l ∈ N there exists γ ∈ PH(l) for which |c(a,H
(l)) −
DH(l)(γ)| ≤ C. But, as it follows from the definitions and from the fact that
I is a homomorphism, DH(l)(γ) does not depend on γ and equals −I(α
−lI(α). This immediately implies that µ(α) = vol (M) · I(α).
Acknowledgements. The origins of this paper lie in our joint work with
P.Biran on the paper [10] – we thank him for fruitful collaboration at an
early stage of this project, as well as for his crucial help with Example 1.17
on Lagrangian spheres in projective hypersurfaces. We also thank him and
O.Cornea for pointing out to us a mistake in the original version of this paper
and helping us with the correction (see Section 8). We thank F. Zapolsky
for his help with the “exotic” monotone Lagrangian torus in S2 × S2 dis-
cussed in Example 1.20. We thank C. Woodward for pointing out to us the
link between the special point in the moment polytope of a symplectic toric
manifold and the Futaki invariant, and E. Shelukhin for useful discussions
on this issue. We are also grateful to V.L. Ginzburg, Y. Karshon, Y. Long,
D. McDuff, M. Pinsonnault, D. Salamon and M. Sodin for useful discussions
and communications. We thank K. Fukaya, H. Ohta and K. Ono, the orga-
nizers of the Conference on Symplectic Topology in Kyoto (February 2006),
M. Harada, Y. Karshon, M. Masuda and T. Panov, the organizers of the Con-
ference on Toric Topology in Osaka (May 2006), O. Cornea, V.L. Ginzburg,
E. Kerman and F. Lalonde, the organizers of the Workshop on Floer theory
(Banff, 2007), and A. Fathi, Y.-G. Oh and C. Viterbo, the organizers of the
AMS Summer Conference on Symplectic Topology and Measure-Preserving
Dynamical Systems (Snowbird, July 2007), for giving us an opportunity to
present a preliminary version of this work and for the superb job they did
in organizing these conferences. Finally, we thank an anonymous referee for
helpful comments and corrections.
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Michael Entov Leonid Polterovich
Department of Mathematics School of Mathematical Sciences
Technion Tel Aviv University
Haifa 32000, Israel Tel Aviv 69978, Israel
entov@math.technion.ac.il polterov@post.tau.ac.il
http://arxiv.org/abs/0709.1127
	Introduction and main results
	Many facets of displaceability
	Preliminaries on quantum homology
	An hierarchy of rigid subsets within Floer theory
	Hamiltonian torus actions
	Super(heavy) monotone Lagrangian submanifolds
	An effect of semi-simplicity
	Discussion and open questions
	Strong displaceability beyond Floer theory? 
	Heavy fibers of Poisson-commutative subspaces
	Detecting stable displaceability
	Preliminaries on Hamiltonian Floer theory
	Valuation on QH* (M)
	Hamiltonian Floer theory
	Conley-Zehnder and Maslov indices
	Spectral numbers
	Partial symplectic quasi-states
	Basic properties of (super)heavy sets
	Products of (super)heavy sets 
	Product formula for spectral invariants
	Decorated Z2-graded complexes
	Reduced Floer and Quantum homology
	Proof of Theorem 5.1
	Proof of algebraic Theorem 5.2 
	Stable non-displaceability of heavy sets
	Analyzing stable stems
	Monotone Lagrangian submanifolds 
	Rigidity of special fibers of Hamiltonian actions
	Calabi and mixed action-Maslov
