Floer theory for Lagrangiansubmanifolds ∗Kaoru Ono
Hokkaido University, Sapporo, Japan
Noncommutativity, IHES
November 15, 2006

∗ Based on the joint work with K. Fukaya (Kyoto), Y.-G. Oh (Madison andSeoul) and H. Ohta (Nagoya)moduli spacesof “geometric objects”(non-linear elliptic PDE)

suitable compactiﬁcationtransversality−−−−−−−−−−→
intersection theory
Kuranishi structure

invariants withcompatibility conditions
algebraic structureassociated to geometry• quantum cohomology, Gromov-Witten invariants (Ruan-Tian, McDuﬀSalamon; Fukaya- —, J. Li-G. Tian, Ruan, Siebert) Li-Tian, Behrend-Fantechi• Floer (co)homology for Hamiltonian systems (Floer, Hofer-Salamon, —; Fukaya- —, G. Liu-G. Tian) with pair-of-pants product (Schwarz, ...)
• Floer (co)homology for Lagrangian intersections (Floer, Oh; FukayaOh-Ohta- —, ﬁltered A∞-algebras, bimodules, etc.)• SFT (Eliashberg-Givental-Hofer), ...
(suitable compactiﬁcation = stable map compactiﬁcation due to Kontsevich)Note. The ﬁrst two cases: intersection theory on homology level
The thrid case: intersection theory on chain level1Symplectic manifolds(M, ω) symplectic manifold of dimension 2n(for simplicity, M is assumed to be closed.)ωdω = 0, ω n 6= 0 everywhereJ
almost complex structure compatible with ω, i.e.,gJ (u, v) = ω(u, Jv) is a Riemannian metric
Locally modeled on symplectic vector space (R2n, ω0) (Darboux).

Basic examples the total space of cotangent space of a manifold, Kähler manifolds, etc.

Fact. Existence, contractiblity of the space of such J’s
c1(M ) = c1(T M, J) ∈ H 2(M ; Z) is well-deﬁned.
2

Basic Concepts
ι:L→M
Lagrangian submanifold (today: only embedding)ι∗ω = 0,dim L =

1
dim M.
2

Examples. • curves on a 2-dimensional symplectic manifold• the graph of a closed 1-form on a C ∞-manifold X
Let Gη be the graph of η ∈ Ω1(X) (a section of T ∗X → X).
Gη ⊂ T ∗X is Lagrangian ⇔ dη = 0• the graph of a symplectomorphism φ ∈ Symp(M, ω)Let φ : M → M be a diﬀeomorphism of M and Γφ its graph. Set(P, Ω) = (M × M, −pr∗1ω + pr∗2ω).
Γφ ⊂ P is Lagrangian ⇔ φ∗ω = ω.
• the real part of projective algebraic manifold deﬁned over R
e.g., RP n ⊂ CP n (the ﬁxed point set of an anti-symplectic involution)
3

The Hamiltonian vector ﬁeld Xh associated to h ∈ C ∞(M ) isdeﬁned byi(Xh)ω = dh.
Note. LXh ω = 0. (Cf. LX ω = 0 ⇔ i(X)ω a closed 1-form)∼ H 1(M ; R).
{X|LX ω = 0}/{Hamiltonian vector ﬁelds} =H = {ht} a smooth family of functions, i.e., H : R × M → R,
ht = H(t, ·),H

Ham v. f.

Ã{Xht }

integrate

Ã

{ϕH
t }

Deﬁnition. ϕ ∈ Diﬀ(M ) is called a Hamiltonian diﬀeomorphismiﬀ ϕ = ϕH
1 for some H = {ht}.
4Arnold’s conjecture for ﬁxed points of Ham diﬀeo
For ϕ ∈ Ham(M, ω), #Fix(ϕ) ≥ min{#Crit(h)|h ∈ C ∞(M )}.
RHS = LS category of M ≥ cup-length (M )∀p ∈ Fix(ϕ) non-deg. ⇒ #Fix(ϕ) ≥ min{#Crit(h)|h Morse}
Morse theory gives lower bound for RHS. (sum of Betti numbers,...)
Note. Compare with the case of diﬀeomorphisms, homeomorphisms.

Analogous question for Lagrangian intersections
L ⊂ M Lagrangian submanifold, ϕ ∈ Ham(M, ω)#L ∩ ϕ(L) ≥ min{#Crit(f )|f ∈ C ∞(L)}?
L transv. φ(L) ⇒ LHS ≥ min{#Crit(f )|f Morse}?
5There are cases when (a weaker version of) estimate hold: great
circles on the round 2-sphere, the zero section of the cotangent
bundle (Laudenbach-Sikorav, Hofer)
In general, no such estimates. (e.g., small circles on the round2-sphere) It is not only a bad news. L ∩ ϕ(L) = ∅ ⇒ Existence of
non-constant pseudo-holomorphic discs with boundary on L ⇒non-existence of exact Lagrangian submanifold in Cn (Gromov),non-degeneracy of Hofer’s distance (Chekanov, Oh, originally
due to Hofer, Lalonde-McDuﬀ), etc.

6

Set-up for Floer theory of Lagrangian intersections
L0, L1 ⊂ M such that L0 t L2
P(L0, L1) : the space of paths from L0 to L1
Deﬁne the closed 1-form α, in a formal sense, on P(L0, L1) by
Z 1

αL0,L1 (ξ) =

0

³

´

ω ξ(t), γ̇(t) dt for ξ ∈ “Tγ P(L0, L1)00

Note that Zero(αL0,L1 ) = L0 ∩ L1

Action functionaleP(L0 , L1)↓
P(L0, L1)

AL0 ,L1→

R

dAL0,L1 = π ∗αL0,L1

7

Maslov-Viterbo index
There are inﬁnitely many positive, resp. negative eigenvalues ofthe Hessian operator of the action functional at critical points.
Although the usual Morse index does not make sense, we canconsider the relative index along paths joining critical points. Ife
we take an appropriate covering space of P(L
0 , L1), the absoluteindex can be deﬁned:µL0,L1 : Crit(AL0,L1 ) → Z

8J = {Jt} a family of compatible alm cplx str ⇒ (formal) L2metric
Formally, gradient ﬂow lines are described as follows.

(Perturbed) Cauchy-Riemann equation∂u + J(u) ∂u = 0∂τ∂t

for u : R × [0, 1] → Mu(τ, i) ∈ Li, i = 0, 1
with suitable asymptotic condition at τ → ±∞.
R ∂u 2For solutions, the energy is given by E(u) = | ∂τ | dτ dt.

Condition for bounded ﬂow lines
E(u) < ∞ ⇔ ∃γ ± =lim u(τ, t)

τ →±∞

9

Floer complex (CF•, δ) CF• is Floer-Novikov completion of thefree module generated by CritA. Grading is given by µ.
δ(x) =

X#M(x, y)y,y:µ(y)=µ(x)+1

where M(x, y) is the moduli space of grad. ﬂow lines from x toy.
Novikov ring ΛL0,L1 .
Floer-Novikov completion of the group ring of the covering transe
formation group of P(L
0 , L1 ) → P(L0 , L1 ).
(CF•, δ) is a “complex” over the Novikov ring.
Orientation issue is non-trivial in the case of Lagrangian intersections.
10

Floer-Novikov completion.
R: ring (later consider R containing Q)
Floer complex

CFq := {+∞X

aixi|ai ∈ R, xi ∈ Crit(A) s.t. (∗)}

i=1(∗) A(xi) → +∞ (i → +∞),µ(xi) = q
Novikov ringΓ: covering transformation group of the Floer-Novikov coveringΛ := {+∞
X

aigi|ai ∈ R, gi ∈ Γ, s.t. (∗∗)}

i=0(∗∗) IA(gi) → +∞ (i → +∞),ewhere IA(gi) = A(gi(x)) − A(x), x ∈ LM
or Pe (L0, L1).
11

Crucial Problem. (1) δ well deﬁned? (2) δ ◦ δ = 0?

Ham systems Floer: monotone case (c1(M ) = λ[ω] for ∃λ > 0),
Hofer-Salamon, —: weakly monotone (semi-positive) case (noJ-holomorphic spheres with negative ﬁrst Chern # for genericJ)∼ •+n(M ).
In these cases, HF•({ϕHt }, J) = H
Lagrangian intersection Floer: π2(M, L) = 0 ⇒ (1) and (2)∼ H•+c(L) with Z/2Z-coeﬀ. Oh:
holds. Moreover, HF•(L, ϕ(L)) =monotone Lagrangian submanifolds L0, L1 with min Maslov # ≥3, monotone Lagrangian submanifold L and ϕH
1 (L) with min
Maslov # ≥ 2 ⇒ (1) and (2) hold.
∃ spectral sequence converging to HF•(L, ϕ(L)). (There arecases when it degenerates at E2-level.)
12

Hamiltonian systems Work with Q-coeﬃcients.
Theorem.(Fukaya- —, Liu-Tian) For any closed symplectic man∼ •+n(M ; Q)⊗
ifold, (1) and (2) holds. Moreover, HF•({ϕHt }, J) = HΛω .
Corollary For ϕ ∈ Ham(M, ω) only with non-deg ﬁxed points,#Fix(ϕ) ≥X

bp(M ).

p13

Lagrangian caseIn general, δ ◦ δ may not be 0. E.g., small circles on the roundsphere.

Obstruction theory (Fukaya-Oh-Ohta- —)By Kuranishi structure, multi-valued perturbation technique, compatible system of orientations (cf. weak spin structure) ⇒ virtual
fundamental chain M(p, q). In contrast to case of Ham systems,
not only bubbling-oﬀ of J-hol spheres, but also J-hol discs (realcodimension-one phenomenon), which may cause δ ◦ δ 6= 0

Systematic study of all hol discs ⇒ obstruction classes to deﬁne
HF, ﬁltered A∞-algebra associated to L.
14•∀ obstruction vanishes (⇔ ∃ sol. of Maurer-Cartan eq. forﬁltered A∞-algebra) ⇒ HF deﬁned by revising δ, Hamiltoinan
invariance (under suitable choice of sol. M-C eq.), ∃ spectral
sequence converging to HF•(L, L).

15

Filtered A∞-algebra associated to L ⊂ M
A∞-algebraC •: graded module. Set C[1]q := C q+1.
k
mk : (C[1]•)⊗ → C[1]• of degree 1 (after shifting degrees)Lk
BC[1] := k (C[1]•)⊗ , bar complex
b on BC[1] and write db = P mbExtend mk to coderivation m
k
k k.
b b
(C •, {mk }∞k=1 ) an A∞-algebra iﬀ d ◦ d = 0.
Note. If mk = 0 for ∀k > 2, DGA (before shifting the degree).
intersection of chains:transversality fails for e.g. self-intersection.
m1 usual boundary operation (up to sign)
To deﬁne m2, take intersection of chains after perturbation ⇒associativity fails in strict sense, but holds up to homotopy.
16

A few formulae
m1 ◦ m1 = 0
m1 ◦ m2 + m2 ◦ (m1 ⊗ id. ± id. ⊗ m1) = 0
m1 ◦ m3 + m2 ◦ (m2 ⊗ id. ± id. ⊗ m2) + m3 ◦ (m1 ⊗ id. ⊗ id. ± id. ⊗ m1 ⊗id. ± id. ⊗ id. ⊗ m1) = 0...
ﬁltered case: 1 ∈ Λ0,nov ⊂ B(C[1] ⊗ Λnov )m1 ◦ m0 = 0m1 ◦ m1 + m2 ◦ (m0(1) ⊗ id. ± id. ⊗ m0(1)) = 0
m1 ◦ m2 + m2 ◦ (m1 ⊗ id. ± id. ⊗ m1) + m3 ◦ (m0(1) ⊗ id. ⊗ id. ± id. ⊗m0(1) ⊗ id. ± id. ⊗ id. ⊗ m0(1)) = 0....
17

C •(L): a suitable countably generated subcomplex of singularchain module with the grading given by codimension of chain(We adopt cohomological convention.)
More precisely, we deal with currents represented by singularchains.
m1(P ) = ±∂P , m2(P1, P2) perturbed intersection of P1 and P2.
mk (P1, . . . , Pk ) using parametrized family of perturbations of the
diagonal depending on P1, . . . , Pk .
Obtain A∞-algebra, which is “homotopy equivalent” to degree
shift of de Rham DGA of L.

18

ﬁltered A∞-algebras
universal Novikov ring e, T free generators, deg e = 2, deg T = 0
PΛnov = { aieni T λi |ai ∈ Q, ni ∈ Z, λi → +∞}P
Λ0,nov = { aieni T λi ∈ Λnov |λi ≥ 0}.

Consider free module over Λ0,nov generated by energy zero elements, its bar complex and take its completion.
Fix {βi = (λi, ni)} such that λi → +∞.
kb k,β its extension as coderivation.
mk,βi : (C[1]•)⊗ → C[1]•, mi(When β = 0, mk,0 = mk . The contribution from β = 0 is “quantum eﬀect” by holomorphic discs. Our ﬁltered A∞-algebra isconsidered as “quantum deformation” of (classical) A∞-algebra.)Pb k,β ⊗ eni T λi degree 1
db = k,βi miHere m0,β 6= 0 only when β = (λ, n) with λ > 0.
(C • ⊗ Λ0,nov , {mk,β }) ﬁltered A∞-algebra iﬀ db ◦ db = 0.
19

Using b ∈ (C[1] ⊗ Λ0,nov )0 with strictly positive energy, i.e.,Pb =bieni T λi , λi > 0, sdeg bi + 2ni = 0 for ∀i, we can deform the operationsP
mbk,β (P1, . . . , Pk ) = mk+`(b, . . . , b, P1, b, . . . , b, Pi, b, . . . , b, Pk , b, . . . , b),
where ` the number of inserted b’s in arbitrary positions.
•{mbk,β } is also ﬁltered A∞-structure.
P

Write eb := b ⊗ · · · ⊗ b.
b b) = 0.
b sol. of Maurer-Cartan equation iﬀ d(eNote. This is equivalent to that mb0,β = 0 for ∀β. Such a b iscalled bounding (co)chain.
P b
b
b
b
It implies that m1 ◦ m1 = 0, where m1 = m1,β . Thus (C[1]• ⊗Λ0,nov , mb1) is a cochain complex.
20

Filtered A∞-algebra associated to LRβ ∈ π2(M, L) By abuse of notation, we write β = ( β ω, µL(β)),
where µL is the Maslov index.
Mk+1(β; J): moduli space of bordered stable maps of genus 0with k + 1-marked points on boundary
Take the ﬁber product by the evaluation maps ev1, . . . , evk :Mk+1(β : P1, . . . , Pk ; J) = Mk+1(β; J) ×L×···×L (P1 × · · · × Pk )Orientation issue, multi-valued perturbation, suitable countable
generated complex ...
Basic idea: Use ev0 : Mk+1(β : P1, . . . , Pk ; J) → L to deﬁne mk.β .
The equation db◦ db = 0 follows from the study of stable compactiﬁcation of moduli spaces of bordered stable maps of genus 0.
(C(L; Λ0,nov )•, {mk,β }) ﬁltered A∞-algebra associated to L. (“homotopy type” is well-deﬁned)21

φ ∈ Symp(M, ω) induces a ﬁltered A∞-morphism φb∗, which is amorphism of coalgebra BC(L, Λ0,nov ))[1] → BC(φ(L), Λ0,nov ))[1]s.t.
bdb ◦ φb∗ = φb∗ ◦ d.

The ﬁltered A∞-algebra depends on choice of J, perturbation,
C •(L), etc. Introduce the notion of homotopy between ﬁltered
A∞-morphisms, then homotopy equivalence. The homotopy
type of the ﬁltered A∞-algebra is uniquely determined by L ⊂ M .

If ∃b ∈ C(L; Λ0,nov )0 sol. of Maurer-Cartan equation, i.e., L is
unobstructed, mb1 ◦ mb1 = 0 ⇒ Floer complex for (L, L) of BottMorse type22

For φ ∈ Symp(M, ω), deﬁne φ∗(b) by
φb∗(eb) = eφ∗(b). Then b: bounding cochain ⇒ φ∗(b): bounding
cochainWe can also deform mk,β using cycles in M (inﬁnitesimal deformation or bulk/boundary deformation). If the Maurer-Cartan
equation has solutions in the ﬁltered A∞-algebra after inﬁnitesimal (or bulk/boundary) deformation, L is called unobstructed
after inﬁnitesimal deformation.
If H•(M ; Q) → H•(L; Q) is surjective, L is unobstructed afterinﬁnitesimal deformation.

23

Filtered A∞-bimodule associated to (L0, L1).
(L0, L1): Lagrangian submanifolds with clean intersection
Construct the ﬁltered A∞-bimodule. (For simiplicity, assume(0)(1)
transversal intersection.) Let Pi
and Pjbe chains in L0 and(0)L1, respectively. Put marked points zi(1)and zjon R × {0, 1}(0)and consider Floer grad. ﬂow lines u such that u(zi
(1)

(1)

(0)

) ∈ Piand u(zj ) ∈ Pj . (After taking stable compactiﬁcation, multivalued perturbation in the sense of Kuranishi structure,) countsuch objects to deﬁne
nk1,k0•••
BC(L1, Λ0,nov ) ⊗CF (L1, L0)⊗BC(L0, Λ0,nov ) → CF •(L1, L0).
•b on BC(L , Λ
Use d’s
i 0,nov ) , i = 1, 0, and nk1,k0⇒ db on BC(L1, Λ0,nov )• ⊗ CF •(L1, L0) ⊗ BC(L0, Λ0,nov )•, which
satisﬁes db ◦ db = 0 ﬁltered A∞-bimodule structure.
24

Extend the coeﬃcient ring from Λ0,nov to Λnov . ϕi ∈ Ham(M, ω)
induces a ﬁltered A∞-bimodule morphism (ϕ1, ϕ0)∗, which is homotopy equivalence of ﬁltered A∞-bimodules.
If ∃bi boundings cochains for Li, i = 0, 1, deform δ tob b1 ⊗ · ⊗ eb0 ) ⇒ Floer complex CF•(L , L ), δ b1,b0 ).
δ b1,b0 (·) = d(e1 0Denote by HF•((L1, b1), (L0, b0). After extending the coeﬃcientring to Λnov , (ϕ1, ϕ0)∗ induces∼ HF•((ϕ (L ), ϕ (b )), (ϕ (L ), ϕ (b ))HF•((L1, b1), (L0, b0) =
1 1
1∗ 1
0 0
0∗ 0

25

Spectral sequence.
•,••,•∃ spectral sequence Er with E1 = CF•(L, L) converging to•,• ∼ •
H (L; Q) ⊗ Λ0,nov .
HF•((L, b)(L, b)), s.t. E2 =26

Some examples of applications.
Theorem. L: Lagrangian submanifold such that H•(M ; Q) →H•(L; Q) is surjective. For ϕ ∈ Ham(M, ω) such that L and ϕ(L)P
are transversal, #(L ∩ ϕ(L)) ≥ bp(L).
Theorem. L: spin Lagrangian submanifold in (R2n, ωcan) suchthat H2(L; Q) = 0. Then the Maslov class µL ∈ H1(L; Z) is
non-zero.
partial aﬃrmative answer to Arnold-Givenal’s conjecture, ...

27

An application to Audin’s conjecture.
Audin’s conjecture Let L be an embedded Lagrangian torus
in the symplectic vector space. Then its minimal Maslov number
is 2.
There are some results due to Polterovich, Viterbo, Oh concerning minimal Maslov number. Also recent works by Fukaya, andCieliebak-Mohnke. Based on our theory, we give a partial result,which is independently obtained by Lev Buhovsky.
Theorem.
Let L be a monotone Lagrangian torus. Then its
minimal Maslov number is 2.
Here monotonicity for L means that µL : π2(M, L) → R andω : π2(M, L) → R are positively proportional.
28

moduli spacesof “geometric objects”(non-linear elliptic PDE)

suitable compactiﬁcationtransversality−−−−−−−−−−→
intersection theory
Kuranishi structure

invariants withcompatibility conditions
algebraic structureassociated to geometry• quantum cohomology, Gromov-Witten invariants (Ruan-Tian, McDuﬀSalamon; Fukaya- —, J. Li-G. Tian, Ruan, Siebert)• Floer (co)homology for Hamiltonian systems (Floer, Hofer-Salamon,—; Fukaya- —, G. Liu-G. Tian) with pair-of-pants product (Schwarz, ...)
• Floer (co)homology for Lagrangian intersections (Floer, Oh; FukayaOh-Ohta- —, ﬁltered A∞-algebras, bimodules, etc.)• SFT (Eliashberg-Givental-Hofer), ...
(suitable compactiﬁcation = stable map compactiﬁcation due to Kontsevich)Note. The ﬁrst two cases: intersection theory on homology level
The thrid case: intersection theory on chain level
29Appendix
Stable maps and compactiﬁcation of moduli spaces
stable maps (Kontsevich)Σ pre-stable curve (at worst nodes) with possibly marked points
Sff
on non-sing part, p : Σ = Σi → Σ normalizationff : Σ → M stable map iﬀ f is continuous, f ◦p is J-hol on each Σiand #Aut(f ) = #{φ| automorphism of Σ s.t. f ◦ φ = f } < ∞.
Fundamental Theorem The moduli space Mg,k (α, J) of stablemaps is a compact Hausdorﬀ space. Here g = genus of Σ, k =#marked points, α ∈ H2(M ; Z).
We have analogous notions and results for grad. ﬂow lines
in Floer theory and for bordered J-holomorphic map with Lagrangian condition.
30

In general, transversality fails, especially at inﬁnity of (uncompactiﬁed) moduli spaces. (e.g., bubbling-oﬀ of multiple cover ofJ-hol sphere with negative c1(M )- number) In order to overcome
this trouble, use Kuranishi structure machinery.

31

Kuranishi structure
A Kuranishi structure on a compact metrizable space X is acompatible system of local models.
(U, E, Γ, s, ψ): Kuranishi neighborhood of x ∈ X
U open subset of Rm, Γ ﬁnite group, E = U × Rk → U Γ-equiv.
vector bundle, s Γ-invariant single-valued section of E → U ,ψ : s−1(0) → X homeo onto its image s.t. x ∈ Imψ.
coordinate change from (U, E, Γ, s, ψ) to (U 0, E 0, Γ0, s0, ψ 0)
Γ → Γ0 injective homomorphism
E ⊂ E0↓↓U ⊂ U0(Γ, Γ0)-equivariant embedding∼ (E 0| )/E(T U 0|U )/T U =U
0(Γ, Γ )-equivariant embeddings is the restriction of s0. rankE −dim U = rankE 0 −dim U 0 Require
certain compatiblity condition for coordinate changes
32

Taking compatible system of multi-valued perturbation of s,
obtain virtual fundamental chain [X]vir over Q from X withoriented Kuranishi structure. (X with Kuranishi structure “without boundary” ⇒ virtual fundamental cycle. )33

Stable maps and compactiﬁcation of moduli spaces
stable maps (Kontsevich)Σ pre-stable curve (at worst nodes) with possibly marked points
f = SΣ
f → Σ normalization
on non-sing part, p : Σiff : Σ → M stable map iﬀ f is continuous, f ◦p is J-hol on each Σiand #Aut(f ) = #{φ| automorphism of Σ s.t. f ◦ φ = f } < ∞.
Fundamental Theorem The moduli space Mg,k (α, J) of stablemaps is a compact Hausdorﬀ space. Here g = genus of Σ, k =#marked points, α ∈ H2(M ; Z).
We have analogous notions and results for grad. ﬂow lines
in Floer theory and for bordered J-holomorphic map with Lagrangian condition.
34

In general, transversality fails at inﬁnity of (uncompactiﬁed)
moduli spaces. (e.g., bubbling-oﬀ of multiple cover of J-hol
sphere with negative c1(M )- number) In order to overcome thistrouble, use Kuranishi structure machinery.

Before compactiﬁcation:∂ J u = 0 (non-linear elliptic PDE) ⇒ Fredholm set-up (approximation by ﬁnite dimensional models) ⇒ Kuranishi neighborhood
stable map compactiﬁcation (#Aut ﬁnite):gluing theorem with index computation, etc ⇒ Kuranishi neighborhood
Obtain a compatible system of Kuranishi neighborhood, i.e., Kuranishi structure.
35

Σ without boundary ⇒ bubbling-oﬀ of hol sphere, degeneration
are real codimension-two or more ⇒ Mg,k (α, J) carries virtual
fundamental cycle⇒ G-W invariants with Kontsevich-Manin’s axiom (except motivic axiom), quantum cohomology and its associativity.

Floer complex for Hamiltonian systems ⇒ [M(p, q)]vir enjoy expected properties (essentially, lack of compactness in codimensionone:broken grad. ﬂow lines) ⇒ HF•(H, J), invariance under Ham
deformations, etc
For computation, pick ht ≡ h C 2-small Morse function. S 1∼ H•+n(M ; Q) ⊗symmetry in Kuranishi structure ⇒ HF•(H, J) =Λω (Similar argument works for {φt} with small ﬂux.)
36