Reduced phase space and toric variety coordinatizations of Delzant
spaces
J.J. Duistermaat∗ and A. Pelayo†
November 4, 2018
Abstract
In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase
space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations.
For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization
of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain
of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex
algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms
of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods
to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and
as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make
the presentation more self-contained.
1 Introduction
Let (M, σ) be a smooth compact and connected symplectic manifold of dimension 2n and let T be a
torus which acts effectively on (M, σ) by means of symplectomorphisms. If the action of T on (M, σ)
is moreover Hamiltonian, then dimT ≤ n, and the image of the momentum mapping µT : M → t
∗ is a
convex polytope ∆ in the dual space t∗ of t, where t denotes the Lie algebra of T . In the maximal case when
dimT = n, (M, σ) is called a Delzant space.
Delzant [3, (*) on p. 323] proved that in this case the polytope ∆ is very special, a so-called Delzant
polytope, of which we recall the definition in Section 2. Furthermore Delzant [3, Th. 2.1] proved that two
Delzant spaces are T -equivariantly symplectomorphic if and only if their momentum mappings have the
same image up to a translation by an element of t∗. Thirdly Delzant [3, pp. 328, 329] proved that for every
Delzant polytope ∆ there exists a Delzant space such that µT (M) = ∆. This Delzant space is obtained as
the reduced phase space for a linear Hamiltonian action of a torus N on a symplectic vector space E, at a
value λN of the momentum mapping of the Hamiltonian N -action, where E, N and λN are determined by
the Delzant polytope.
Finally Delzant [3, Sec. 5] observed that the Delzant polytope gives rise to a fan (= éventail in French),
and that the Delzant space with Delzant polytope ∆ is T -equivariantly diffeomorphic to the toric variety
Mtoric defined by the fan. HereMtoric is a complex n-dimensional complex analytic manifold, and the action
of the real torus T on Mtoric has an extension to a complex analytic action on Mtoric of the complexification
∗Research stimulated by a KNAW professorship
†Partly supported by a Rackham Predoctoral fellowship
http://arxiv.org/abs/0704.0430v1
TC of T . In our description in Section 5 of the toric variety M
toric we do not use fans. The information, for
each vertex v of ∆, which codimension one faces of ∆ contain v, already suffices to define Mtoric.
In this note we show that the construction of the Delzant space M as a reduced phase space leads, for
every vertex v of the Delzant polytope, to a natural coordinatization ϕv of a T -invariant open cell Mv in
M , where Mv contains the unique fixed point mv in M of the T -action such that µT (mv) = v. We give an
explicit construction of the inverse ψv of ϕv, which is a maximal diffeomorphism in the sense of Remark
3.10. The construction of ψv originated in an attempt to extend the equivariant symplectic ball embeddings
from (B2nr , σ0) ⊂ (C
n, σ0) into the Delzant space (M, σ) in Pelayo [10] by maximal equivariant symplec-
tomorphisms from open neighborhoods of the origin in Cn into the Delzant space (M, σ). If v and w are
two different vertices, then the coordinate transformation ϕw ◦ ϕv
−1 is given by the explicit formulas (4.3),
(4.4).
Let Σ be the set of all strata of the orbit type stratification of M for the T -action. Then the domain of
definition Mv of ϕv is equal to the union of all S ∈ Σ such that the fixed point mv belongs to the closure
of S in M , see Corollary 5.5. The strata S ∈ Σ are also the orbits in the toric variety Mtoric ≃ M for the
action of the complexification TC of the real torus T , and the domain of definition Mv of ϕv is equal to
the domain of definition of a natural complex analytic TC-equivariant coordinatization Φv of a TC-invariant
open cell. The diffeomorphism Φv ◦ ϕv
−1, which sends the reduced phase space coordinates to the toric
variety coordinates, maps Uv := ϕv(Mv) diffeomorphically onto a complex vector space, and is given by
the explicit formulas (5.9).
In the toric variety coordinates the complex structure is the standard one and the coordinate transfor-
mations are relatively simple Laurent monomial transformations, whereas the symplectic form is generally
given by quite complicated algebraic functions. On the other hand, in the reduced phase space coordinates
the symplectic form is the standard one, but the coordinate transformations, and also the complex structure,
have a more complicated appearance.
Let F denote the set of all d codimension one faces of ∆ and, for every vertex v of ∆, let Fv denote the
set of all f ∈ F such that v ∈ f . Note that #(Fv) = n for every vertex v of ∆. For any sets A and B, let
AB denote the set of all A-valued functions on B. If A is a field and the set B is finite, then AB is a #(B)-
-dimensional vector space over A. One of the technical points in this paper is the efficient organization of
proofs and formulas made possible by viewing the Delzant space as a reduction of the vector space CF , and
letting, for each vertex v, the coordinatizations ϕv and Φv take their values in C
Fv . This leads to a natural
projection ρv : C
F → CFv obtained by the restriction of functions on F to Fv ⊂ F . For each vertex v the
complex vector space CFv is isomorphic to Cn, but the isomorphism depends on an enumeration of Fv , the
introduction of which would lead to an unnecessary complication of the combinatorics. Similarly our torus
T is isomorphic to Rn/Zn, but the isomorphism depends on the choice of a Z-basis of the integral lattice
tZ in the Lie algebra t of T . As for each vertex v a different Z-basis of tZ appears, we also avoid such a
choice, keeping T in its abstract form. We hope and trust that this will not lead to confusion with our main
references Delzant [3], Audin [2] and Guillemin [7] about Delzant spaces, where CF , each CFv , and T is
denoted as Cd, Cn, and Rn/Zn, respectively.
The organization of this manuscript is as follows. In Section 2 we review the definition of the reduced
phase Delzant space, and introduce the notations which will be convenient for our purposes. In Section 3 we
define the reduced phase space coordinatizations. In Section 4 we give explicit formulas for the coordinate
transformations and describe the reduced phase space Delzant space as obtained by gluing together bounded
open subsets of n-dimensional complex vector spaces with these coordinate transformations as the gluing
maps. In Section 5 we review the definition of the toric variety defined by the Delzant polytope, prove that
the natural mapping from the reduced phase space to the toric variety is a diffeomorphism, and compare
the coordinatizations of Section 3 with the natural coordinatizations of the toric variety. In Section 6 we
present these computations for the two simplest classes of examples, the complex projective spaces and the
Hirzebruch surfaces.
2 The reduced phase space
Let T be an n-dimensional torus, a compact, connected, commutative n-dimensional real Lie group, with
Lie algebra t. It follows that the exponential mapping exp : t → T is a surjective homomorphism from
the additive Lie group t onto T . Furthermore, tZ := ker(exp) is a discrete subgroup of (t, +) such that the
exponential mapping induces an isomorphism from t/tZ onto T , which we also denote by exp. Note that tZ
is defined in terms of the group T rather than only the Lie algebra t, but the notation tZ has the advantage
over the more precise notation TZ that it reminds us of the fact it is a subgroup of the additive group t.
Because t/tZ is compact, tZ has a Z-basis which at the same time is an R-basis of t, and each Z-basis
of tZ is an R-basis of t. Using coordinates with respect to an ordered Z-basis of tZ, we obtain a linear
isomorphism from t onto Rn which maps tZ onto Z
n, and therefore induces an isomorphism from T onto
n/Zn. For this reason, tZ is called the integral lattice in t. However, because we do not have a preferred
Z-basis of tZ, we do not write T = R
n/Zn.
Let ∆ be an n-dimensional convex polytope in t∗. We denote by F and V the set of all codimension one
faces and vertices of ∆, respectively. Note that, as a face is defined as the set of points of the closed convex
set on which a given linear functional attains its minimum, see Rockafellar [11, p.162], every face of ∆ is
compact. For every v ∈ V , we write
Fv = {f ∈ F | v ∈ f}.
∆ is called a Delzant polytope if it has the following properties, see Guillemin [7, p. 8].
i) For each f ∈ F there is an Xf ∈ t and λf ∈ R such that the hyperplane which contains f is equal to
the set of all ξ ∈ t∗ such that 〈Xf , ξ〉 + λf = 0, and ∆ is contained in the set of all ξ ∈ t
∗ such that
〈Xf , ξ〉+ λf ≥ 0. The vector Xf and constant λf are made unique by requiring that they are not an
integral multiple of another such vector and constant, respectively.
ii) For every v ∈ V , the Xf with f ∈ Fv form a Z-basis of the integral lattice tZ in t.
It follows that
∆ = {ξ ∈ t∗ | 〈Xf , ξ〉+ λf ≥ 0 for every f ∈ F}. (2.1)
Also, #(Fv) = n for every v ∈ V , which already makes the polytope ∆ quite special. In the sequel we
assume that ∆ is a given Delzant polytope in t∗.
For any z ∈ CF and f ∈ F we write z(f) = zf , which we view as the coordinate of the vector z with
the index f . Let π be the real linear map from RF to t defined by
π(t) :=
tf Xf , t ∈ R
F . (2.2)
Because, for any vertex v, the Xf with f ∈ Fv form a Z-basis of tZ which is also an R-basis of t, we have
π(ZF ) = tZ and π(R
F ) = t. It follows that π induces a surjective homomorphism of Lie groups π′ from
the torus RF/ZF = (R/Z)F onto t/tZ, and we have the corresponding surjective homomorphism exp ◦π
from RF /ZF onto T .
Write n := ker π, a linear subspace of RF , and N = ker(exp ◦π′), a compact commutative subgroup
of the torus RF /ZF . Actually, N is connected, see Lemma 3.1 below, and therefore isomorphic to n/nZ,
where nZ := n ∩ Z
F is the integral lattice in n of the torus N . 1
On the complex vector space CF of all complex-valued functions on F we have the action of the torus
F/ZF , where t ∈ RF/ZF maps z ∈ CF to the element t · z ∈ CF defined by
(t · z)f = e
2π i tf zf , f ∈ F.
The infinitesimal action of Y ∈ RF =Lie(RF /ZF ) is given by
(Y · z)f = 2π i Yf zf ,
which is a Hamiltonian vector field defined by the function
z 7→ 〈Y, µ(z)〉 =
Yf |zf |
2/2 =
Yf (xf
2 + yf
2)/2, (2.3)
and with respect to the symplectic form
σ := (i /4π)
dzf ∧dzf = (1/2π)
dxf ∧dyf , (2.4)
if zf = xf +i yf , with xf , yf ∈ R. Here the factor 1/2π is introduced in order to avoid an integral lattice
(2π Z)F instead of our ZF .
Because the right hand side of (2.3) depends linearly on Y , we can view µ(z) as an element of (RF )∗ ≃
F , with the coordinates
µ(z)f = |zf |
2/2 = (xf
2 + yf
2)/2, f ∈ F. (2.5)
In other words, the action of RF/ZF on CF is Hamiltonian, with respect to the symplectic form σ and with
momentum mapping µ : CF → (Lie(RF /ZF ))∗ given by (2.3), or equivalently (2.5).
It follows that the subtorus N of RF /ZF acts on CF in a Hamiltonian fashion, with momentum mapping
µN := ι
◦ µ : CF → n∗, (2.6)
where ιn : n → R
F denotes the identity viewed as a linear mapping from n ⊂ RF to RF , and its transposed
: (RF )∗ → n∗ is the map which assigns to each linear form on RF its restriction to n.
Write λN = ι
(λ), where λ denotes the element of (RF )∗ ≃ RF with the coordinates λf , f ∈ F .
It follows from Guillemin [7, Th. 1.6 and Th. 1.4] that λN is a regular value of µN , hence the level set
Z := µN
−1({λN}) of µN for the level λN is a smooth submanifold of C
F , and that the action of N on Z
is proper and free. As a consequence the N -orbit space M = M∆ := Z/N is a smooth 2n-dimensional
manifold such that the projection p : Z → M exhibits Z as a principal N -bundle over M . Moreover, there
is a unique symplectic form σM on M such that p
∗σM = ιZ
∗σ, where ιZ is the identity viewed as a smooth
mapping from Z to CF .
Remark 2.1 Guillemin [7] used the momentum mapping µN − λN instead of µN , such that the reduction
is taken at the zero level of his momentum mapping. We follow Audin [2, Ch. VI, Sec. 3.1] in that we use
the momentum mapping µN for the N -action, which does not depend on λ, and do the reduction at the level
λN . ⊘
1We did not find a proof of the connectedness of N in [3], [2], or [7].
The symplectic manifold (M, σM ) is the Marsden-Weinstein reduction of the symplectic manifold
(CF , σ) for the Hamiltonian N -action at the level λN of the momentum mapping, as defined in Abraham
and Marsden [1, Sec. 4.3]. On the N -orbit space M , we still have the action of the torus (RF /ZF )/N ≃ T ,
with momentum mapping µT :M → t
∗ determined by
π∗ ◦ µT ◦ p = (µ− λ)|Z . (2.7)
The torus T acts effectively on M and µT (M) = ∆, see Guillemin [7, Th. 1.7]. Actually, all these
properties of the reduction will also follow in a simple way from our description in Section 3 of Z in term
of the coordinates zf , f ∈ F .
The symplectic manifold M∆ together with this Hamiltonian T -action is called the Delzant space de-
fined by ∆, see Guillemin, [7, p. 13]. This proves the existence part [3, pp. 328, 329] of Delzant’s theory.
3 The reduced phase space coordinatizations.
For any v ∈ V , let ιv := ρv
∗ : (RFv)∗ → (RF )∗ denote the transposed of the restriction projection
ρv : R
F → RFv . If in the usual way we identify (RFv)∗ and (RF )∗ with RFv and RF , respectively, then
ιv : R
Fv → RF is the embedding defined by ιv(x)f = xf if f ∈ Fv and ιv(x)f ′ = 0 if f
′ ∈ F , f ′ /∈ Fv.
Because ιv maps Z
Fv into ZF and ιv(R
Fv) ∩ ZF = ιv(Z
Fv), it induces an embedding of the n-dimensional
torus RFv/ZFv into RF/ZF , which we also denote by ιv.
Lemma 3.1 With these notations, RF , ZF , and RF/ZF are the direct sum of n and ιv(RFv), n ∩ Zn and
Fv), and N and ιv(R
Fv/ZFv), respectively.
It follows that N is connected, a torus, with integral lattice equal to n ∩ ZF . It also follows that π ◦ ιv
is an isomorphism from the torus RFv/ZFv onto the torus T .
Proof Let t ∈ RF . Because the Xf , f ∈ Fv, form an R-basis of t, there exists a unique t
v ∈ RFv , such
π(t) =
(tv)f Xf = π(ιv(t
that is, t− ιv(t
v) ∈ n. Moreover, because the Xf , f ∈ Fv , also form a Z-basis of tZ, we have that t
v ∈ ZFv ,
and therefore t− ιv(t
v) ∈ n ∩ ZF , if t ∈ ZF . �
Lemma 3.2 We have z ∈ Z if and only if µ(z) − λ ∈ π∗(t∗). More explicitly, if and only if there exists a
ξ ∈ t∗ such that
|zf |
2/2− λf = 〈Xf , ξ〉 for every f ∈ F. (3.1)
When z ∈ Z , the ξ in (3.1) is uniquely determined.
Furthermore, Z = (µ− λ)−1(π∗(∆)), (µ− λ)(Z) = π∗(∆), and Z is a compact subset of CF .
Proof The kernel of ι∗
is equal to the space of all linear forms on RF which vanish on n := kerπ, and
therefore ker ι∗
is equal to the image of π∗ : t∗ → (RF )∗. Because π is surjective, π∗ is injective, which
proves the uniqueness of ξ.
It follows from (3.1) that 〈Xf , ξ〉 + λf ≥ 0 for every f ∈ F , and therefore ξ ∈ ∆ in view of (2.1).
Conversely, if ξ ∈ ∆, then there exists for every f ∈ F a complex number zf such that |zf |
2/2 = 〈Xf , ξ〉+
λf , which means that z ∈ Z and (µ− λ)(z) = π
∗(ξ). The set π∗(∆) is compact because ∆ is compact and
π∗ is continuous. Because the mapping µ− λ is proper, it follows that Z = (µ− λ)−1(π∗(∆)) is compact.
Let v ∈ V . The Xf , f ∈ Fv , form an R-basis of t, and therefore there exists for each z ∈ C
Fv a unique
ξ = µv(z) ∈ t
∗ such that (3.1) holds for every f ∈ Fv . That is, the mapping µv : C
Fv → t∗ is defined by
the equations
|zf |
2/2 − λf = 〈Xf , µv(z)〉, z ∈ C
Fv , f ∈ Fv. (3.2)
In other words, µv is defined by the formula
ρv ◦ π
∗ ◦ µv = ρv ◦ (µ− λ) ◦ ιv, (3.3)
where ρv denotes the restriction projection from R
F onto RFv .
Lemma 3.3 If we let T act on CFv via RFv/ZFv by means of (t, z) 7→ (π ◦ ιv)−1(t) ·z, then µv : CFv → t∗
is a momentum mapping for this Hamiltonian action of T on CFv , with µv(0) = v. Here the symplectic
form on CFv is equal to
σ := (i /4π)
dzf ∧dzf = (1/2π)
dxf ∧dyf , (3.4)
that is, (2.4) with F replaced by Fv .
Let ρv denote the restriction projection from C
F onto CFv , and let Uv be the interior of the subset ρv(Z)
of CFv . Write
∆v := ∆ \
f ′∈F\Fv
f ′. (3.5)
Then ρv(Z) = µv
−1(∆), µv(ρv(Z)) = ∆, Uv = µv
−1(∆v), and µv(Uv) = ∆v. In particular ρv(Z) is a
compact subset of CFv , and Uv is a bounded and connected open neighborhood of 0 in C
Proof The first statement follows from (3.3), the fact that ρv◦µ◦ιv is a momentum mapping for the standard
Fv/ZFv action on CFv , and the fact that a momentum mapping for a Hamiltonian action plus a constant is a
momentum mapping for the same Hamiltonian action. It follows in view of (3.2) that 〈Xf , µv(0)〉+λf = 0
for every f ∈ Fv , hence µv(0) = v in view of i) in the definition of a Delzant polytope, and the fact that {v}
is the intersection of all the f ∈ Fv .
It follows from (3.2), Lemma 3.2, that z ∈ Z if and only if
|zf |
2/2 = 〈Xf , µv(ρv(z))〉 + λf for every f ∈ F, (3.6)
where we note that these equations are satisfied by definition for the f ∈ Fv. Therefore, if z ∈ Z , then (3.6)
and (2.1) imply that µv(ρ(z)) ∈ ∆. Conversely, if ξ ∈ ∆, then it follows from Lemma 3.2 that there exists
z ∈ Z such that π∗(ξ) = µ(z)− λ, of which the restriction to Fv yields ξ = µv(ρ(z)).
If ξ ∈ ∆v, z
v ∈ CFv , µv(z
v) = ξ, then 〈Xf ′ , µv(z
v)〉 + λf ′ > 0 for every f
′ ∈ F \ Fv, which will
remain valid if we replace zv by z̃v in a sufficiently small neighborhood of zv in CFv . It follows that we can
find z̃ ∈ CF such that ρv(z̃) = z̃
v and (3.6) holds with z replaced by z̃. That is, z̃ ∈ Z , and we have proved
that zv ∈ Uv.
Let conversely z ∈ Uv ⊂ C
Fv . We have in view of (3.2) that
|zf |
2/2 = 〈Xf , µv(z)− µv(0)〉 = 〈Xf , µv(z)− v〉
for every f ∈ Fv. Therefore µv(z) − v is multiplied by c
2 if we replace z by c z, c > 0. Because z is in
the interior of ρv(Z), we have c z
v ∈ ρv(Z), hence µv(c z) ∈ ∆ for c > 1, c sufficiently close to 1. On the
other hand, if ξ belongs to a face of ∆ which is not adjacent to v, then v + τ (ξ − v) /∈ ∆ for any τ > 1. It
follows that µv(z) does not belong to any f
′ ∈ F \ Fv, that is, µv(z) ∈ ∆v. �
The equation (3.6) can be written in the form |zf | = rf (µv(ρv(z))), where, for each f ∈ F , the function
rf : ∆ → R≥0 is defined by
rf (ξ) := (2(〈Xf , ξ〉+ λf ))
1/2, f ∈ F, ξ ∈ ∆. (3.7)
We now view the equations (3.6) for z ∈ Z as equations for the coordinates zf ′ , f
′ ∈ F \ Fv, with the
zf , f ∈ Fv as parameters, where the latter constitute the vector z
v = ρv(z). If z
v ∈ Uv, then for each
f ′ ∈ F \ Fv the coordinate zf ′ lies on the circle about the origin with strictly positive radius rf ′(µv(z
Because Lemma 3.1 implies that the homomorphism which assigns to each element of N its projection to
F\Fv/ZF\Fv is an isomorphism, and the latter torus is the group of the coordinatewise rotations of the zf ′ ,
f ′ ∈ F \ Fv, this leads to the following conclusions.
Proposition 3.4 Let v be a vertex of ∆. The open subset Zv := ρv−1(Uv) ∩ Z of Z is a connected smooth
submanfold of CF of real dimension 2n + (d − n), where d = #(F ) and d − n = dimN . The action
of the torus N on Zv is free, and the projection ρv : Zv → Uv exhibits Zv as a principal N -bundle over
Uv. It follows that we have a reduced phase space Mv := Zv/N , which is a connected smooth symplectic
2n-dimensional manifold, which carries an effective Hamiltonian T -action with momentum mapping as in
(2.7), with Z replaced by Zv.
There is a unique global section sv : Uv → Zv of ρv : Zv → Uv such that sv(z)f ′ ∈ R>0 for every
z ∈ Uv and f
′ ∈ F \ Fv. Actually, sv(z)f ′ = rf ′(µv(z)) when z ∈ Uv and f
′ ∈ F \ Fv , and therefore the
section sv is smooth. If pv : Zv → Mv = Zv/N denotes the canonical projection, then ψv := pv ◦ sv is a
T -equivariant symplectomorphism from Uv onto Mv, where T acts on Uv via R
Fv/ZFv , as in Lemma 3.3.
Remark 3.5 When z belongs to the closure ρv(Z) = Uv of Uv in CFv , see Lemma 3.3, we can define
sv(z) ∈ C
F by sv(z)f = zf when f ∈ Fv and sv(z)f ′ = rf ′(µv(z)) when f
′ ∈ F \ Fv . This defines
a continuous extension sv : Uv → C
F of the mapping sv : Uv → Z . Therefore sv(Uv) ⊂ Z , and
ψv := p ◦ sv : Uv →M is a continuous extension of the diffeomorphism ψv : Uv →Mv .
The continuous mapping ψv : Uv → M is surjective, but the restriction of it to the boundary ∂Uv :=
Uv \Uv of Uv in C
Fv is not injective. If zv ∈ ∂Uv, then the set G of all f
′ ∈ F \ Fv such that sv(z
v)f ′ = 0,
or equivalently µT (ψv(z
v)) ∈ f ′, is not empty. The fiber of ψv over ψv(z
v) is equal to the set of all tv · zv,
where the tv ∈ RFv/ZFv are of the form
tvf = −
(v)fg tg, f ∈ Fv,
where tg ∈ R/Z. It follows that each fiber is an orbit of some subtorus of R
Fv/ZFv acting on CFv . ⊘
Recall the definition (3.5) of the open subset ∆v of the Delzant polytope ∆. Because the union over all
vertices v of the ∆v is equal to ∆, we have the following corollary.
Corollary 3.6 The sets Zv, v ∈ V , form a covering of Z . As a consequence, Z is a smooth submanifold
of CF of real dimension n + d. The action of the torus N on Z is free, and we have a reduced phase
space M := Z/N , which is a compact and connected smooth 2n-dimensional symplectic manifold, which
carries an effective Hamiltonian T -action with momentum mapping µT :M → T as in (2.7). The sets Mv,
v ∈ V , form an open covering of M and the ϕv := (ψv)
−1 : Mv → Uv form an atlas of T -equivariant
symplectic coordinatizations of the Hamiltonian T -space M . For each v ∈ V , we have Mv = µT
−1(∆v),
and µT |Mv = µv ◦ ϕv.
For a characterization of Mv in terms of the orbit type stratification in M for the T -action, see Corollary
5.5, which also implies that Mv is an open cell in M .
Corollary 3.7 For every f ∈ F the set µT−1(f) is a real codimension two smooth compact connected
smooth symplectic submanifold of M .
For each v ∈ V , the set Mv is dense in M , and the diffeomorphism ψv : Uv → Mv is maximal among
all diffeomorphisms from open subsets of CFv onto open subsets of M .
Proof If f ∈ F , then for each v ∈ V we have that
−1(f) = {z ∈ Uv | zf = 0} (3.8)
if v ∈ f , that is, f ∈ ∆v. This follows from (3.2) and i) in the description of ∆ in the beginning of Section
2. On the other hand, µv
−1(f) = ∅ if f /∈ ∆v. Because µT
−1(f) ∩Mv = ψv(µv
−1(f)), and the Mv,
v ∈ V , form an open covering of M , this proves the first statement. The second statement follows from the
first one, because the complement of Mv in M is equal to the union of the sets µT
−1(f ′) with f ′ ∈ F \ Fv.
Remark 3.8 It follows from the proof of Corollary 3.7, that µT−1(f) is a connected component of the
fixed point set in M of the of the circle subgroup exp(RXf ) of T .
Actually, µT
−1(f) is a Delzant space for the action of the (n− 1)-dimensional torus
T/exp(RXf ),
with Delzant polytope P ⊂ (t/(RXf ))
∗ such that the image of P in t∗ under the embedding (t/(RXf ))
∗ is equal to a translate of f .
In a similar way, if g is a k-dimensional face of ∆, then µT
−1(g) is a 2k-dimensional Delzant space for
the quotient of T by the subtorus of T which acts trivially on µT
−1(g). ⊘
Remark 3.9 Let, for each f ∈ F , cf ∈H
2(M, Z) ⊂H2(M, R) denote the Poincaré dual of the codimen-
sion two Delzant subspace µT
−1(f) of M , see Remark 3.8. Then, with our normalization of the symplectic
form (2.4), the de Rham cohomology class [σM ] of the symplectic form σM of the Delzant space is equal to
[σM ] =
λf cf , (3.9)
see Guillemin [8, Thm. 6.3]. In particular [σM ] ∈H
2(M, Z), and therefore [σM ] is equal to the Chern class
of a complex line bundle over M , if all the coefficients λf , f ∈ F , are integers.
If ∆ is a simplex, when M is isomorphic to the n-dimensional complex projective space, then the
−1(f), f ∈ F , are complex projective hyperplanes, see Subsection 6.1, which are all homologous to
each other. It follows that in this case [σM ] = γ c, where c is the Poincaré dual of a complex projective
hyperplane and γ is equal to the sum of all the coefficients λf , f ∈ F . ⊘
Remark 3.10 Let ι : T → Rn/Zn be an isomorphism of tori, which allows us to let t ∈ T act on Cn via
n/Zn by means of
(t · z)j = e
2π i ι(t)j zj, 1 ≤ j ≤ n.
Let U be a connected T -invariant open neighborhood of 0 in Cn, provided with the symplectic form (2.4)
with F replaced by {1, . . . , n}. Let ψ : U → M be a T -equivariant symplectomorphism from U onto an
open subset ψ(U) of M . Because 0 is the unique fixed point for the T -action in U , and the fixed points
for the T -action in M are the pre-images under µT of the vertices of ∆, there is a unique v ∈ V such
that µT (ψ(0)) = v. Let Iv : C
Fv → Cn denote the complex linear extension of the tangent map of the
torus isomorphism ι ◦ (π ◦ ιv). In terms of the notation of Lemma 3.3 and Proposition 3.4, we have that
U ⊂ Iv(Uv) and ψv = ψ ◦ Iv on Iv
−1(U), which leads to an identification of ψ with the restriction of ψv to
the connected open subset Iv
−1(U) of Uv, via the isomorphism Iv
The ψ’s, with U equal to a ball in Cn centered at the origin, are the equivariant symplectic ball embed-
dings in Pelayo [10], and the second statement in Corollary 3.7 shows that the diffeomorphisms ψv are the
maximal extensions of these equivariant symplectic ball embeddings. ⊘
4 The coordinate transformations
Let v, w ∈ V . Then
Uv, w := ϕv(Mv ∩Mw) = Uv ∩ ψv
−1 ◦ ψw(Uw)
= {zv ∈ Uv | (z
v)f 6= 0 for every f ∈ Fv \ Fw}. (4.1)
In this section we will give an explicit formula for the coordinate transformations
ϕw ◦ ϕv
−1 = ψw
−1 ◦ ψv : Uv, w → Uw, v,
which then leads to a description of the Delzant space M as obtained by gluing together the subsets Uv with
the coordinate transformations as the gluing maps.
Let f ∈ F . Because the Xg , g ∈ Fw, form a Z-basis of tZ, and Xf ∈ tZ, there exist unique integers
f , g ∈ Fw, such that
f Xg. (4.2)
Note that if f ∈ Fw, then (w)
f = 1 when g = f and (w)
f = 0 otherwise. For the following lemma recall
that rg is defined by expression (3.7).
Lemma 4.1 Let v, w ∈ V , zv ∈ Uv, w. Then zw := ϕw ◦ ϕv−1(zv) ∈ Uw ⊂ CFw is given by
zwg =
(zvf )
f∈Fv\Fw
|zvf |
f (4.3)
if g ∈ Fw ∩ Fv, and
zwg =
(zvf )
f rg(µv(z
f∈Fv\Fw
|zvf |
f (4.4)
if g ∈ Fw \ Fv.
Proof The element zw ∈ Uw is determined by the condition that sw(zw) belongs to the N -orbit of sv(zv).
That is,
w)f = e
i tf sv(z
v)f for every f ∈ F
for some t ∈ RF such that ∑
tf Xf = 0. (4.5)
It follows from (4.5), (4.2) and the linear independence of the Xg, g ∈ Fw, that t ∈ n if and only if
tg = −
f∈F\Fw
tf for every g ∈ Fw. (4.6)
Note that µv(z
v) = µT (m) = µw(z
w), where m = ψv(z
v) = ψw(z
w). It follows from the definition of
the sections sv and sw, see Proposition 3.4, that
i) sv(z
v)f = z
f and sw(z
w)f = z
f if f ∈ Fv ∩ Fw,
ii) sv(z
v)f = z
f and sw(z
w)f = rf (µw(z
w)) = rf (µv(z
v)) if f ∈ Fv \ Fw,
iii) sv(z
v)f = rf (µv(z
v)) and sw(z
w)f = z
f if f ∈ Fw \ Fv, and
iv) sv(z
v)f = rf (µv(z
v)) = rf (µw(z
w)) = sw(z
w)f if f ∈ F \ (Fv ∪ Fw).
It follows from ii) and iv) that tf = −arg z
f and tf = 0 modulo 2π if f ∈ Fv \Fw and f ∈ F \ (Fv ∪Fw),
respectively. Then (4.6) implies that, modulo 2π,
f∈Fv\Fw
f arg z
f for every g ∈ Fw.
It now follows from i) and iii) that if g ∈ Fw, then z
g = sw(z
w)g = e
i tg sv(z
v)g is equal to
ei tg zvg =
(zvf )
f∈Fv\Fw
|zvf |
if g ∈ Fv, and equal to
ei tg |zvg | =
(zvf )
f |zvg |/
f∈Fv\Fw
|zvf |
if g /∈ Fv , respectively. Here we have used that if g ∈ Fw, then (w)
= 1 if f = g and (w)g
= 0 if f ∈ Fw,
f 6= g. Because |zvg | = rg(µv(z
v)) if g /∈ Fv, see (3.6) and (3.7), this completes the proof of the lemma. �
Remark 4.2 Note that zv ∈ Uv, w means that zv ∈ Uv and zvf 6= 0 if f ∈ Fv \ Fw. Furthermore, z
v ∈ Uv
implies that if g /∈ Fv , then µv(z
v) /∈ g, and therefore rg is smooth on a neighborhood of µv(z
v). Finally,
note that if g ∈ Fw and f ∈ Fv ∩Fw, then (w)
∈ {0, 1}, and therefore each of the factors in the right hand
sides of (4.3) and (4.4) is smooth on Uv, w. ⊘
Remark 4.3 In (4.3) and (4.4) only the integers (w)gf appear with f ∈ Fv and g ∈ Fw. Let (w v) denote the
matrix (w)gf , where f ∈ Fv and g ∈ Fw. Then (w v) is invertible, with inverse equal to the integral matrix
(v w). These integral matrices also satisfy the cocycle condition that (w v) (v u) = (w u), if u, v, w ∈ V .
These properties follow from the fact that (4.2) shows that (w v) is the matrix which maps the Z-basis Xg,
g ∈ Fw, onto the Z-basis Xf , f ∈ Fv, of tZ. It is no surprise that these base changes enter in the formulas
which relate the models in the vector spaces CFv for the different choices of v ∈ V . ⊘
Corollary 4.4 Let, for each v ∈ V , the mapping µv : CFv → t∗ be defined by (3.2), which is a momentum
mapping for a Hamiltonian T -action via RFv/ZFv on the symplectic vector space CFv as in Lemma 3.3.
Define Uv := µv
−1(∆v). If also w ∈ V , define Uv, w as the right hand side of (4.1), and, if z
v ∈ Uv, w,
define ϕw, v(z
v) := zw, where zw ∈ CFw is given by (4.3) and (4.4).
Then ϕw, v is a T -equivariant symplectomorphism from Uv, w onto Uw, v such that µw = µv ◦ ϕv, w
on Uw, v. The ϕw, v satisfy the cocycle condition ϕw, v ◦ ϕv, u = ϕw, u where the left hand side is defined.
Glueing together the Hamiltonian T -spaces Uv, v ∈ V , with the momentum maps µv, by means of the
gluing maps ϕw, v, v, w ∈ V , we obtain a compact connected smooth symplectic manifold M̃ with an
effective Hamiltonian T -action with a common momentum map µ̃ : M̃ → T such that µ̃(M̃ ) = ∆. In other
words, M̃ is a Delzant space for the Delzant polytope ∆.
The Delzant space M̃ is obviously isomorphic to the Delzant spaceM = µ−1({λ})/N introduced in Section
2, and actually the isomorphism is used in the proof that M̃ is a Delzant space for the Delzant polytope ∆.
The only purpose of Corollary 4.4 is to exhibit the Delzant space as obtained from gluing together the Uv,
v ∈ V , by means of the gluing maps ϕv, w, v, w ∈ V .
5 The toric variety
Let T := {z ∈ C | |z| = 1} denote the unit circle in the complex plane. The mapping t 7→ u where
uf = e
2π i tf for every f ∈ F is an isomorphism from the torus RF/ZF onto TF , where TF acts on CF by
means of coordinatewise multiplication and RF/ZF acted on CF via the isomorphism from RF/ZF onto
F . The complexification TC of the compact Lie group T is the multiplicative group C
× of all nonzero
complex numbers, and the complexification of TF is equal to TF
:= (TC)
F = (C×)F , which also acts on
F by means of coordinatewise multiplication.
The complexification NC of N is the subgroup exp(nC) of U
, where nC := n⊕ i n ⊂ C
F denotes the
complexification of n, viewed as a complex linear subspace, a complex Lie subalgebra, of the Lie algebra
F of TF
. In view of (4.6), we have, for every v ∈ V , that NC is equal to the set of all t ∈ T
such that
f∈F\Fv
f , g ∈ Fv. (5.1)
This implies that NC is a closed subgroup of T
isomorphic to T
, and therefore NC is a reductive
complex algebraic group.
If we define
v := {z ∈ C
F | zf 6= 0 for every f ∈ F \ Fv}, (5.2)
then it follows from (5.1) that the action of NC on C
v is free and proper. It follows that the action of NC on
v (5.3)
is free and proper, and therefore the NC-orbit space
Mtoric := CF∆/NC (5.4)
has a unique structure of a complex analytic manifold of complex dimension n such that the canonical pro-
jection from CF∆ onto M
toric exhibits CF∆ as a principal NC-bundle over M
toric. On Mtoric we still have the
complex analytic action of the complex Lie group group TF
/NC, which is isomorphic to the complexifi-
cation TC of our real torus T induced by the projection π. The complex analytic manifold M
toric together
with the complex analytic action of TC on it is the toric variety defined by the polytope ∆ in the title of this
section.
If v ∈ V and z ∈ CFv , then it follows from (5.1) that there is a unique t ∈ NC such that tf = zf for
every f ∈ F \ Fv, or in other words, z = t · ζ , where ζ ∈ C
F is such that ζf = 1 for every f ∈ F \ Fv.
Let Sv : C
Fv → CFv be defined by Sv(z
v)f = z
f when f ∈ F and Sv(z
v) = 1 when f ∈ F \ Fv, as in
Audin [2, p. 159]. If Pv : C
v → C
v /NC denotes the canonical projection from C
v onto the open subset
Mtoricv := C
v /NC of M
toric, then Ψv := Pv ◦ Sv is a complex analytic diffeomorphism from C
Fv onto
Mtoricv . It is TC-equivariant if we let TC act on C
Fv via TFv
as in Lemma 3.3. We use the diffeomorphism
Φv := Ψv
−1 from Mtoricv onto C
Fv as a coordinatization of the open subset Mtoricv of M
toric.
If v, w ∈ V , then
Utoricv, w := Φv(M
toric
toric
w ) = C
Fv ∩Ψv
−1 ◦Ψw(C
= {zv ∈ CFv | (zv)f 6= 0 for every f ∈ Fv \ Fw}. (5.5)
Moreover, with a similar argument as for Lemma 4.1, actually much simpler, we have that for every zv ∈
Utoricv, w the element z
w := Φw ◦Φv
−1(zv) ∈ CFw is given by
zwg =
(zvf )
f , g ∈ Fw. (5.6)
In this way the coordinate transformation Φw◦Φv
−1 is a Laurent monomial mapping, much simpler than the
coordinate transformation (4.3), (4.4). It follows that the toric variety Mtoric can be alternatively described
as obtained by gluing the n-dimensional complex vector spaces CFv , v ∈ V , together, with the maps (5.6)
as the gluing maps. This is the kind of toric varieties as introduced by Demazure [4, Sec. 4].
For later use we mention the following observation of Danilov [5, Th. 9.1], which is also of interest in
itself.
Lemma 5.1 Mtoric is simply connected.
Proof Let w ∈ V . It follows from (5.5), for all v ∈ V , that the complement of Mtoricw in Mtoric is equal to
the union of finitely closed complex analytic submanifolds of complex codimension one, whereas Mtoricw is
contractible because it is diffeomorphic to the complex vector space CFw . Because complex codimension
one is real codimension two, any loop in Mtoric with base point in Mtoricw can be slightly deformed to such
a loop which avoids the complement of Mtoricw in Mtoric, that is, which is contained in M
toric
w , after which it
can be contracted within Mtoricw to the base point in M
toric
w . �
Recall the definition in Section 2 of the reduced phase space M = Z/N .
Theorem 5.2 The identity mapping from Z into CF∆, followed by the canonical projection P from C
Mtoric = CF∆/NC, induces a T -equivariant diffeomorphism ̟ from M = Z/N onto M
toric. It follows that
each NC-orbit in C
∆ intersects Z in an N -orbit in Z .
Proof Because N is a closed Lie subgroup of NC, we have that the mapping P : Z → CF∆/NC induces a
mapping ̟ : Z/N → CF∆/NC, which moreover is smooth.
If v ∈ V , z ∈ Zv, then it follows from (5.1) that the tf , f ∈ F \ Fv , of an element t ∈ NC can take
arbitrary values, and therefore the |zf |, f ∈ F \ Fv can be moved arbitrarily by means of infinitesimal
NC-actions. Because Z is defined by prescribing the |zf |, f ∈ F \ Fv, as a smooth function of the zf ,
f ∈ Fv, and the Zv, v ∈ V , form an open covering of Z , this shows that at each point of Z the NC-orbit is
transversal to Z , which implies that ̟ is a submersion.
It follows that ̟(M) is an open subset of Mtoric. Because M is compact and ̟ is continuous, ̟(M)
is compact, and therefore a closed subset of Mtoric. Because Mtoric is connected, the conclusion is that
̟(M) =Mtoric, that is, ̟ is surjective.
Because ̟ is a surjective submersion, dimRM = 2n =dimRM
toric, and M is connected, we conclude
that ̟ is a covering map. Because Mtoric is simply connected, see Lemma 5.1, we conclude that ̟ is
injective, that is, ̟ is a diffeomorphism. �
Remark 5.3 Theorem 5.2 is the last statement in Delzant [3], with no further details of the proof. Audin
[2, Prop. 3.1.1] gave a proof using gradient flows, whereas the injectivity has been proved in [7, Sec. A1.2]
using the principle that the gradient of a strictly convex function defines an injective mapping. ⊘
Note that in the definition of the toric variety Mtoric, the real numbers λf , f ∈ F , did not enter, whereas
these numbers certainly enter in the definition of M , the symplectic form on M , and the diffeomorphism
̟ :M →Mtoric. Therefore the symplectic form σtoricλ := (̟
−1)∗(σ) on Mtoric on Mtoric will depend on the
choice of λ ∈ RF . On the symplectic manifold (Mtoric, σtoricλ ), the action of the maximal compact subgroup
T of TC is Hamiltonian, with momentum mapping equal to
µtoricλ := µ ◦̟
−1 :Mtoric → t∗, (5.7)
where µtoricλ (M
toric) = ∆, where we note that ∆ in (2.1) depends on λ.
In the following lemma we compare the reduced phase space coordinatizations with the toric variety
coordinatizations.
Lemma 5.4 Let v ∈ V . Then Mtoricv = ̟(Mv), and
θv := Ψv
−1 ◦̟ ◦ ψv (5.8)
is a TFv-equivariant diffeomorphism from Uv onto C
For each zv ∈ Uv, the element ζ
v := θv(z
v) is given in terms of zv by
ζvf = z
f ′∈F\Fv
rf ′(µv(z
f ′ , f ∈ Fv , (5.9)
where the functions rf ′ : ∆ → R≥0 are given by (3.7). We have
µv(zv) = µT (ψv(z
v)) = µtoricλ (Ψv(ζ
v)), (5.10)
and zv = θv
−1(ζv) is given in terms of ζ
zvf = ζ
f ′∈F\Fv
rf ′(ξ)
f ′ , f ∈ Fv, (5.11)
where ξ is the element of ∆ equal to the right hand side of (5.10).
Proof It follows from Lemma 3.3 and the paragraph preceding Proposition 3.4 that if zv ∈ ρv(Z), then
zv ∈ Uv if and only if z
f ′ 6= 0 for every f
′ ∈ F \ Fv. That is, the set Zv in Proposition 3.4 is equal to
Z ∩ CFv . It therefore follows from Theorem 5.2 that each NC-orbit in the NC-invariant subset C
v of C
intersects the N -invariant subset Zv of Z in an N -orbit in Zv, that is,
Mtoricv = Pv(C
v ) = ̟(pv(Zv)) = ̟(Mv).
If zv ∈ Uv, then Proposition 3.4 implies that sv(z
v)f = z
f for every f ∈ Fv and
v)f ′ = rf ′(µv(z
v)), f ′ ∈ F \ Fv.
If we define t ∈ TF
tf ′ = rf ′(µv(z
v))−1, f ′ ∈ F \ Fv,
f ′∈F\Fv
rf ′(µv(z
f ′ , f ∈ Fv ,
then (t · sv(z
v))t′ = 1 for every t
′ ∈ F \ Fv and, for every f ∈ Fv, ζ
f := (t · sv)f is equal to the right hand
side of (5.9). That is, t · sv(z
v) = Sv(ζ
v), see the definition of Sv in the paragraph preceding (5.5). On the
other hand, it follows from (5.1) that t ∈ NC, and therefore
Ψv(ζv) = Pv(t · sv(z
v)) = Pv(sv(z
v)) = ̟ ◦ pv(sv(z
v)) = ̟ ◦ ψv(z
that is, ζv = Ψv
−1 ◦̟ ◦ ψv(z
v). �
Corollary 5.5 Let s be the relative interior of a face of ∆. Then µT−1(s) is equal to a stratum S of the
orbit type stratification in M of the T -action, and also equal to the preimage under ̟ : M → Mtoric of a
TC-orbit in M
toric. If s = {v} for a vertex v, then µT
−1(s) = {mv} for the unique fixed point mv in M for
the T -action such that µT (mv) = v.
The mapping s 7→ µT
−1(s) is a bijection from the set Σ∆ of all relative interiors of faces of ∆ onto the
set Σ of all strata of the orbit type stratificiation in M for the action of T . If s, s′ ∈ Σ∆ then s is contained
in the closure of s′ in ∆ if and only if µT
−1(s) is contained in the closure of µT
−1(s′) in M .
The domain of definition Mv of ϕv in M is equal to the union of the S ∈ Σ such that mv belongs
to the closure of S in M . The domain of definition Mtoricv = ̟(Mv) of Φv is equal to the union of the
corresponding strata of the T -action in Mtoric, each of which is a TC-orbit in M
toric. Mv and M
toric
v are
open cells in M and Mtoric, respectively.
Proof There exists a vertex v of ∆ such that v belongs to the closure of s in t∗, which implies that s is
disjoint from all f ′ ∈ F \ Fv. Let Fv, s denote the set of all f ∈ Fv such that s ⊂ f , where Fv, s = ∅ if and
only if s is the interior of ∆. For any subset G of Fv, let C
G denote the set of all z ∈ C
Fv such that zf = 0
if f ∈ G and zf 6= 0 if f ∈ Fv \G. It follows from µv = µT ◦ ψv and (3.8) that ψ
v (µT
−1(s)) is equal to
Uv ∩ C
G with G = Fv, s. The diffeomorphism θv maps this set onto the set C
G with G = Fv, s. Because
the sets of the form CFvG with G ⊂ Fv are the strata of the orbit type stratification of the T
Fv -action on CFv ,
and also equal to the (TC)
Fv -orbits in CFv , the first statement of the corollary follows.
The second statement follows from µv
−1({v}) = {0} and the fact that 0 is the unique fixed point of the
Fv-action in Uv.
If s ∈ Σ∆ and v ∈ V , then mv belongs to the closure of µT
−1(s) if and only if s is not contained in any
f ′ ∈ F \ Fv. This proves the characterization of the domain of definition Mv := Zv/N = µT
−1(∆v) of
ϕv. The last statement follows from the fact that Φv is a diffeomorphism from M
toric
v onto the vector space
Fv , and ̟ is a diffeomorphism from Mv onto M
toric
v . �
Remark 5.6 If v, w ∈ V , then
ϕw ◦ ϕv
−1 = ψw
−1 ◦ ψv = θw
−1 ◦ (Ψw
−1 ◦Ψv) ◦ θv = θw
−1 ◦ (Φw ◦Φv
−1) ◦ θv.
Using the formula (5.6) for Φw ◦ Φv
−1, this can be used in order to obtain the formulas (4.3), (4.4) as a
consequence of (5.9). In the proof, it is used that ξ := µv(z
v) = µw(z
w), |zvf | = rf (ξ) if f ∈ Fv \ Fw, and
= (w)
if f ′ ∈ F \ Fv and g ∈ Fw. ⊘
In the following corollary we describe the symplectic form σtoricλ on the toric variety M
toric in the toric
variety coordinates.
Corollary 5.7 For each v ∈ V , the symplectic form (Φv
−1)∗(σtoricλ ) on C
Fv is equal to (θv
−1)∗(σv), where
σv is the standard symplectic form on C
Fv given by (3.4).
Because rf ′(µv(z
v))2 is an inhomogeneous linear function of the quantities |zvf |
2, it follows from (5.9) that
the equations which determine the |zvf |
2 in terms of the quantities |ζvf |
2 are n polyomial equations for the n
unknowns |zfv |2, f ∈ Fv , where the coefficients of the polyomials are inhomogeneous linear functions of the
|ζvf |, f ∈ Fv. In this sense the |z
2, f ∈ Fv, are algebraic functions of the |ζ
2, f ∈ Fv, and substituting
these in (5.9) we obtain that the diffeomorphism θv
−1 from CFv onto Uv is an algebraic mapping. If ∆ is a
simplex, when Mtoric is the n-dimensional complex projective space, we have an explicit formula for θv
see Subsection 6.1. However, already in the case that ∆ is a planar quadrangle, when Mtoric is a complex
two-dimensional Hirzebruch surface, we do not have an explicit formula for θv
−1. See Subsection 6.2.
Summarizing, we can say that in the toric variety coordinates the complex structure is the standard
one and the coordinate transformations are the relatively simple Laurent monomial transformations (5.6).
However, in the toric variety coordinates the λ-dependent symplectic form in general is given by quite
complicated algebraic functions. On the other hand, in the reduced phase space coordinates the symplectic
form is the standard one, but the coordinate transformations (4.3), (4.4) are more complicated. Also the
complex structure in the reduced phase space coordinates, which depends on λ, is given by more complicated
formulas.
Remark 5.8 It is a challenge to compare the formula in Corollary 5.7 for the symplectic form in toric
variety coordinates with Guillemin’s formula in [7, Th. 3.5 on p. 141] and [8, (1.3)]. Note that in the
latter the pullback by means of the momentum mapping appears of a function on the interior of ∆, where in
general we do not have a really explicit formula for the momentum mapping in toric variety coordinates. ⊘
6 Examples
6.1 The complex projective space
Let ∆ be an n-dimensional simplex in t∗. A little bit of puzzling shows that there is a Z-basis ei, 1 ≤ i ≤ n,
of the integral lattice tZ in t, such that, with the notation
e0 = −
ei, (6.1)
the Xf , f ∈ F , are the ei, 0 ≤ i ≤ n. That is, in the sequel we write F = {0, 1, . . . , n}. The Delzant
simplex (2.1) is determined by the inequalities 〈ei, ξ〉+ λi ≥ 0, 0 ≤ i ≤ n, which has a non-empty interior
if and only if
λi > 0. (6.2)
In the sequel we take for v the vertex determined by the equations 〈ei, ξ〉+ λi = 0 for all 1 ≤ i ≤ n, where
Fv = {1, . . . , n}. If we write ξi = 〈ei, ξ〉, 1 ≤ i ≤ n, when ξ ∈ t
∗, then (3.2) yields that
v)i = |zi|
2/2− λi, 1 ≤ i ≤ n.
It follows from (3.7) that
r0(ξ) = (2(−
ξi + λ0))
and therefore (5.9) yields that
ζvi = z
i (2γ − ‖z
v‖2)−1/2, 1 ≤ i ≤ n, (6.3)
where we have written
‖zv‖2 =
|zvi |
Note that Uv is the open ball in C
n with center at the origin and radius equal to (2c)1/2.
The equations (6.3) imply that
‖ζv‖2 = ‖zv‖
2/(2γ − ‖zv‖2),
hence
‖zv‖2 = 2γ ‖ζv‖2/(1 + ‖ζv‖2).
Therefore the mapping θv−1 : ζv 7→ zv is given by the explicit formulas
zvi = ζ
i (2γ/(1 + ‖ζ
v‖2))1/2, 1 ≤ i ≤ n. (6.4)
It can be verified that the symplectic form (θv
−1)∗(σv), where
σv = (1/2π)
dxvi ∧dy
is the standard symplectic form in (3.4), is equal to γ times the Fubini-Study form in Griffiths and Harris
[6, p. 30, 31]. In view of Remark 3.9 this agrees with the fact that the de Rham cohomology class of the
Fubini-Study form is Poincaré dual to the homology class of a complex projective hyperplane in the complex
projective space, see Griffiths and Harris [6, p. 122].
6.2 The Hirzebruch surface
Let n = 2 and let ∆ be a quadrangle in the t∗ plane. A little bit of puzzling shows that there is an m ∈ Z≥0
and a Z-basis e1, e2 of the integral lattice tZ in t, such that the Xf , f ∈ F , are the ei, 1 ≤ i ≤ 4, with
e3 = −e1 +me2, and e4 = −e2. We recognize the toric variety M
toric as the Hirzebruch surface Σm, see
Hirzebruch [9].
The Delzant polytope (2.1) is determined by the inequalities 〈ei, ξ〉 + λi ≥ 0, 1 ≤ i ≤ 4, which is a
quadrangle if and only if
γ± := λ1 + λ3 ±mλ4 > 0, (6.5)
which inequalities imply that λ2 + λ4 = γ+ + γ− > 0.
In the sequel we take for v the vertex determined by the equations 〈ei, ξ〉+ λi = 0 for i = 1, 2, where
Fv = {1, 2}. If we write ξi = 〈ei, ξ〉, 1 ≤ i ≤ 2, when ξ ∈ t
∗, then (3.2) yields that
v)i = |zi|
2/2 − λi, 1 ≤ i ≤ 2.
It follows from (3.7) that
r3(ξ) = (2(−ξ1 +mξ2 + λ3))
1/2, r4(ξ) = (2(−ξ2 + λ4))
and therefore (5.9) yields that
ζv1 = z
1 (2γ−|z
2 +m |zv2 |
2)−1/2, (6.6)
ζv2 = z
2 (2γ−|z
2 +m |zv2 |
2)m/2 (2(γ+ + γ−)− |z
2)−1/2. (6.7)
If we write ti = |z
2 and τi = |ζ
2, then this leads to the equations
τ1 = t1/(2γ− − t1 +mt2),
τ2 = t2 (2γ− + t1 +mt2)
m/(2(γ+ + γ−)− t2)
for t1, t2. If we solve t1 from the first equation,
t1 = (2γ− +mt2) τ1/(1 + τ1),
and substitute this into the second equation, then this leads to the polynomial equation
(1 + τ1)
m τ2 (2(γ+ + γ−)− t2) = t2 (2γ− +mt2)
m (6.8)
of degree m + 1 for t2. If we substract the left hand side from the right hand side then the derivative with
respect to t2 is strictly positive, and one readily obtains that for every τ1, τ2 ∈ R≥0 there is a unique solution
t2 ∈ R≥0, confirming the first statement in Lemma 5.4.
On the other hand, if we work over C, and view both the parameter ε := (1 + τ1)
m τ2 and the unknown
t2 as elements of the complex projective line P
1, then the equation (6.8) defines a complex algebraic curve
C in the (t2, ε)-plane P
1 × P1, where the restriction to C of the projection to the first variable t2 is a
complex analytic diffeomorphism from C onto P1, as on C we have that ε is a complex analytic function
of t2. In particular C is irreducible. The restriction to C of the projection to the second variable ε is an
(m+ 1)-fold branched covering. Over ε = 0 and over ε = ∞ we have that m of the m+ 1 branches come
together, whereas there are two more branch points on the ε-line over which only two of the branches come
together. The fact that C is irreducible implies that the part of C over the complement of the branch points is
connected, and therefore the analytic continuation of any solution t2 of (6.8), as a complex analytic function
of ε in the complement of the branch points, will reach each other branch if ε runs over a suitable loop. In
other words, the solution t2 is an algebraic function of ε of degree m+ 1, and no branch of a solution is of
lower degree. This holds in particular for our solutions t2 ∈ R≥0 for ε ∈ R≥0.
References
[1] R. Abraham and J.E. Marsden: Foundations of Mechanics. Benjamin/Cummings Publ. Co., London,
etc., 1978.
[2] M. Audin: The Topology of Torus Actions on Symplectic Manifolds. Birkhäuser, Basel, Boston, Berlin,
1991.
[3] T. Delzant: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math.
France 116 (1988) 315–339.
[4] M. Demazure: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. scient. Éc.
Norm. Sup. 3 (1970) 507–588.
[5] V.I. Danilov: The geometry of toric varieties. Russ. Math. Surveys 33:2 (1978) 97–154, translated from
Uspekhi Mat. Nauk SSSR 33:2 (1978) 85–134.
[6] P. Griffiths and J. Harris: Principles of Algebraic Geometry. J. Wiley & Sons, Inc., New York, etc.,
1978.
[7] V. Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces. Birkhäuser,
Boston, etc., 1994.
[8] V. Guillemin: Kaehler structures on toric varieties. J. Differential Geometry 40 (1994) 285–309.
[9] F. Hirzebruch: Über eine Klasse von einfach-zusammenhn̈genden komplexen Mannigfaltigkeiten.
Mathematische Annalen 124 (1951) 77–86.
[10] A. Pelayo: Topology of spaces of equivariant symplectic embeddings. Proc. Amer. Math. Soc. 135
(2007) 277–288.
[11] R.T. Rockafellar: Convex Analysis. Princeton University Press, princeton, N.J., 1970.
J.J. Duistermaat
Mathematisch Instituut, Universiteit Utrecht
P.O. Box 80 010, 3508 TA Utrecht, The Netherlands
e-mail: duis@math.uu.nl
A. Pelayo
Department of Mathematics, University of Michigan
2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA
e-mail: apelayo@umich.edu
	Introduction
	The reduced phase space
	The reduced phase space coordinatizations.
	The coordinate transformations
	The toric variety
	Examples
	The complex projective space
	The Hirzebruch surface

  In this note we describe the natural coordinatizations of a Delzant space
defined as a reduced phase space (symplectic geometry view-point) and give
explicit formulas for the coordinate transformations. For each fixed point of
the torus action on the Delzant polytope, we have a maximal coordinatization of
an open cell in the Delzant space which contains the fixed point. This cell is
equal to the domain of definition of one of the natural coordinatizations of
the Delzant space as a toric variety (complex algebraic geometry view-point),
and we give an explicit formula for the toric variety coordinates in terms of
the reduced phase space coordinates. We use considerations in the maximal
coordinate neighborhoods to give simple proofs of some of the basic facts about
the Delzant space, as a reduced phase space, and as a toric variety. These can
be viewed as a first application of the coordinatizations, and serve to make
the presentation more self-contained.

Introduction
Let (M, σ) be a smooth compact and connected symplectic manifold of dimension 2n and let T be a
torus which acts effectively on (M, σ) by means of symplectomorphisms. If the action of T on (M, σ)
is moreover Hamiltonian, then dimT ≤ n, and the image of the momentum mapping µT : M → t
∗ is a
convex polytope ∆ in the dual space t∗ of t, where t denotes the Lie algebra of T . In the maximal case when
dimT = n, (M, σ) is called a Delzant space.
Delzant [3, (*) on p. 323] proved that in this case the polytope ∆ is very special, a so-called Delzant
polytope, of which we recall the definition in Section 2. Furthermore Delzant [3, Th. 2.1] proved that two
Delzant spaces are T -equivariantly symplectomorphic if and only if their momentum mappings have the
same image up to a translation by an element of t∗. Thirdly Delzant [3, pp. 328, 329] proved that for every
Delzant polytope ∆ there exists a Delzant space such that µT (M) = ∆. This Delzant space is obtained as
the reduced phase space for a linear Hamiltonian action of a torus N on a symplectic vector space E, at a
value λN of the momentum mapping of the Hamiltonian N -action, where E, N and λN are determined by
the Delzant polytope.
Finally Delzant [3, Sec. 5] observed that the Delzant polytope gives rise to a fan (= éventail in French),
and that the Delzant space with Delzant polytope ∆ is T -equivariantly diffeomorphic to the toric variety
Mtoric defined by the fan. HereMtoric is a complex n-dimensional complex analytic manifold, and the action
of the real torus T on Mtoric has an extension to a complex analytic action on Mtoric of the complexification
∗Research stimulated by a KNAW professorship
†Partly supported by a Rackham Predoctoral fellowship
http://arxiv.org/abs/0704.0430v1
TC of T . In our description in Section 5 of the toric variety M
toric we do not use fans. The information, for
each vertex v of ∆, which codimension one faces of ∆ contain v, already suffices to define Mtoric.
In this note we show that the construction of the Delzant space M as a reduced phase space leads, for
every vertex v of the Delzant polytope, to a natural coordinatization ϕv of a T -invariant open cell Mv in
M , where Mv contains the unique fixed point mv in M of the T -action such that µT (mv) = v. We give an
explicit construction of the inverse ψv of ϕv, which is a maximal diffeomorphism in the sense of Remark
3.10. The construction of ψv originated in an attempt to extend the equivariant symplectic ball embeddings
from (B2nr , σ0) ⊂ (C
n, σ0) into the Delzant space (M, σ) in Pelayo [10] by maximal equivariant symplec-
tomorphisms from open neighborhoods of the origin in Cn into the Delzant space (M, σ). If v and w are
two different vertices, then the coordinate transformation ϕw ◦ ϕv
−1 is given by the explicit formulas (4.3),
(4.4).
Let Σ be the set of all strata of the orbit type stratification of M for the T -action. Then the domain of
definition Mv of ϕv is equal to the union of all S ∈ Σ such that the fixed point mv belongs to the closure
of S in M , see Corollary 5.5. The strata S ∈ Σ are also the orbits in the toric variety Mtoric ≃ M for the
action of the complexification TC of the real torus T , and the domain of definition Mv of ϕv is equal to
the domain of definition of a natural complex analytic TC-equivariant coordinatization Φv of a TC-invariant
open cell. The diffeomorphism Φv ◦ ϕv
−1, which sends the reduced phase space coordinates to the toric
variety coordinates, maps Uv := ϕv(Mv) diffeomorphically onto a complex vector space, and is given by
the explicit formulas (5.9).
In the toric variety coordinates the complex structure is the standard one and the coordinate transfor-
mations are relatively simple Laurent monomial transformations, whereas the symplectic form is generally
given by quite complicated algebraic functions. On the other hand, in the reduced phase space coordinates
the symplectic form is the standard one, but the coordinate transformations, and also the complex structure,
have a more complicated appearance.
Let F denote the set of all d codimension one faces of ∆ and, for every vertex v of ∆, let Fv denote the
set of all f ∈ F such that v ∈ f . Note that #(Fv) = n for every vertex v of ∆. For any sets A and B, let
AB denote the set of all A-valued functions on B. If A is a field and the set B is finite, then AB is a #(B)-
-dimensional vector space over A. One of the technical points in this paper is the efficient organization of
proofs and formulas made possible by viewing the Delzant space as a reduction of the vector space CF , and
letting, for each vertex v, the coordinatizations ϕv and Φv take their values in C
Fv . This leads to a natural
projection ρv : C
F → CFv obtained by the restriction of functions on F to Fv ⊂ F . For each vertex v the
complex vector space CFv is isomorphic to Cn, but the isomorphism depends on an enumeration of Fv , the
introduction of which would lead to an unnecessary complication of the combinatorics. Similarly our torus
T is isomorphic to Rn/Zn, but the isomorphism depends on the choice of a Z-basis of the integral lattice
tZ in the Lie algebra t of T . As for each vertex v a different Z-basis of tZ appears, we also avoid such a
choice, keeping T in its abstract form. We hope and trust that this will not lead to confusion with our main
references Delzant [3], Audin [2] and Guillemin [7] about Delzant spaces, where CF , each CFv , and T is
denoted as Cd, Cn, and Rn/Zn, respectively.
The organization of this manuscript is as follows. In Section 2 we review the definition of the reduced
phase Delzant space, and introduce the notations which will be convenient for our purposes. In Section 3 we
define the reduced phase space coordinatizations. In Section 4 we give explicit formulas for the coordinate
transformations and describe the reduced phase space Delzant space as obtained by gluing together bounded
open subsets of n-dimensional complex vector spaces with these coordinate transformations as the gluing
maps. In Section 5 we review the definition of the toric variety defined by the Delzant polytope, prove that
the natural mapping from the reduced phase space to the toric variety is a diffeomorphism, and compare
the coordinatizations of Section 3 with the natural coordinatizations of the toric variety. In Section 6 we
present these computations for the two simplest classes of examples, the complex projective spaces and the
Hirzebruch surfaces.
2 The reduced phase space
Let T be an n-dimensional torus, a compact, connected, commutative n-dimensional real Lie group, with
Lie algebra t. It follows that the exponential mapping exp : t → T is a surjective homomorphism from
the additive Lie group t onto T . Furthermore, tZ := ker(exp) is a discrete subgroup of (t, +) such that the
exponential mapping induces an isomorphism from t/tZ onto T , which we also denote by exp. Note that tZ
is defined in terms of the group T rather than only the Lie algebra t, but the notation tZ has the advantage
over the more precise notation TZ that it reminds us of the fact it is a subgroup of the additive group t.
Because t/tZ is compact, tZ has a Z-basis which at the same time is an R-basis of t, and each Z-basis
of tZ is an R-basis of t. Using coordinates with respect to an ordered Z-basis of tZ, we obtain a linear
isomorphism from t onto Rn which maps tZ onto Z
n, and therefore induces an isomorphism from T onto
n/Zn. For this reason, tZ is called the integral lattice in t. However, because we do not have a preferred
Z-basis of tZ, we do not write T = R
n/Zn.
Let ∆ be an n-dimensional convex polytope in t∗. We denote by F and V the set of all codimension one
faces and vertices of ∆, respectively. Note that, as a face is defined as the set of points of the closed convex
set on which a given linear functional attains its minimum, see Rockafellar [11, p.162], every face of ∆ is
compact. For every v ∈ V , we write
Fv = {f ∈ F | v ∈ f}.
∆ is called a Delzant polytope if it has the following properties, see Guillemin [7, p. 8].
i) For each f ∈ F there is an Xf ∈ t and λf ∈ R such that the hyperplane which contains f is equal to
the set of all ξ ∈ t∗ such that 〈Xf , ξ〉 + λf = 0, and ∆ is contained in the set of all ξ ∈ t
∗ such that
〈Xf , ξ〉+ λf ≥ 0. The vector Xf and constant λf are made unique by requiring that they are not an
integral multiple of another such vector and constant, respectively.
ii) For every v ∈ V , the Xf with f ∈ Fv form a Z-basis of the integral lattice tZ in t.
It follows that
∆ = {ξ ∈ t∗ | 〈Xf , ξ〉+ λf ≥ 0 for every f ∈ F}. (2.1)
Also, #(Fv) = n for every v ∈ V , which already makes the polytope ∆ quite special. In the sequel we
assume that ∆ is a given Delzant polytope in t∗.
For any z ∈ CF and f ∈ F we write z(f) = zf , which we view as the coordinate of the vector z with
the index f . Let π be the real linear map from RF to t defined by
π(t) :=
tf Xf , t ∈ R
F . (2.2)
Because, for any vertex v, the Xf with f ∈ Fv form a Z-basis of tZ which is also an R-basis of t, we have
π(ZF ) = tZ and π(R
F ) = t. It follows that π induces a surjective homomorphism of Lie groups π′ from
the torus RF/ZF = (R/Z)F onto t/tZ, and we have the corresponding surjective homomorphism exp ◦π
from RF /ZF onto T .
Write n := ker π, a linear subspace of RF , and N = ker(exp ◦π′), a compact commutative subgroup
of the torus RF /ZF . Actually, N is connected, see Lemma 3.1 below, and therefore isomorphic to n/nZ,
where nZ := n ∩ Z
F is the integral lattice in n of the torus N . 1
On the complex vector space CF of all complex-valued functions on F we have the action of the torus
F/ZF , where t ∈ RF/ZF maps z ∈ CF to the element t · z ∈ CF defined by
(t · z)f = e
2π i tf zf , f ∈ F.
The infinitesimal action of Y ∈ RF =Lie(RF /ZF ) is given by
(Y · z)f = 2π i Yf zf ,
which is a Hamiltonian vector field defined by the function
z 7→ 〈Y, µ(z)〉 =
Yf |zf |
2/2 =
Yf (xf
2 + yf
2)/2, (2.3)
and with respect to the symplectic form
σ := (i /4π)
dzf ∧dzf = (1/2π)
dxf ∧dyf , (2.4)
if zf = xf +i yf , with xf , yf ∈ R. Here the factor 1/2π is introduced in order to avoid an integral lattice
(2π Z)F instead of our ZF .
Because the right hand side of (2.3) depends linearly on Y , we can view µ(z) as an element of (RF )∗ ≃
F , with the coordinates
µ(z)f = |zf |
2/2 = (xf
2 + yf
2)/2, f ∈ F. (2.5)
In other words, the action of RF/ZF on CF is Hamiltonian, with respect to the symplectic form σ and with
momentum mapping µ : CF → (Lie(RF /ZF ))∗ given by (2.3), or equivalently (2.5).
It follows that the subtorus N of RF /ZF acts on CF in a Hamiltonian fashion, with momentum mapping
µN := ι
◦ µ : CF → n∗, (2.6)
where ιn : n → R
F denotes the identity viewed as a linear mapping from n ⊂ RF to RF , and its transposed
: (RF )∗ → n∗ is the map which assigns to each linear form on RF its restriction to n.
Write λN = ι
(λ), where λ denotes the element of (RF )∗ ≃ RF with the coordinates λf , f ∈ F .
It follows from Guillemin [7, Th. 1.6 and Th. 1.4] that λN is a regular value of µN , hence the level set
Z := µN
−1({λN}) of µN for the level λN is a smooth submanifold of C
F , and that the action of N on Z
is proper and free. As a consequence the N -orbit space M = M∆ := Z/N is a smooth 2n-dimensional
manifold such that the projection p : Z → M exhibits Z as a principal N -bundle over M . Moreover, there
is a unique symplectic form σM on M such that p
∗σM = ιZ
∗σ, where ιZ is the identity viewed as a smooth
mapping from Z to CF .
Remark 2.1 Guillemin [7] used the momentum mapping µN − λN instead of µN , such that the reduction
is taken at the zero level of his momentum mapping. We follow Audin [2, Ch. VI, Sec. 3.1] in that we use
the momentum mapping µN for the N -action, which does not depend on λ, and do the reduction at the level
λN . ⊘
1We did not find a proof of the connectedness of N in [3], [2], or [7].
The symplectic manifold (M, σM ) is the Marsden-Weinstein reduction of the symplectic manifold
(CF , σ) for the Hamiltonian N -action at the level λN of the momentum mapping, as defined in Abraham
and Marsden [1, Sec. 4.3]. On the N -orbit space M , we still have the action of the torus (RF /ZF )/N ≃ T ,
with momentum mapping µT :M → t
∗ determined by
π∗ ◦ µT ◦ p = (µ− λ)|Z . (2.7)
The torus T acts effectively on M and µT (M) = ∆, see Guillemin [7, Th. 1.7]. Actually, all these
properties of the reduction will also follow in a simple way from our description in Section 3 of Z in term
of the coordinates zf , f ∈ F .
The symplectic manifold M∆ together with this Hamiltonian T -action is called the Delzant space de-
fined by ∆, see Guillemin, [7, p. 13]. This proves the existence part [3, pp. 328, 329] of Delzant’s theory.
3 The reduced phase space coordinatizations.
For any v ∈ V , let ιv := ρv
∗ : (RFv)∗ → (RF )∗ denote the transposed of the restriction projection
ρv : R
F → RFv . If in the usual way we identify (RFv)∗ and (RF )∗ with RFv and RF , respectively, then
ιv : R
Fv → RF is the embedding defined by ιv(x)f = xf if f ∈ Fv and ιv(x)f ′ = 0 if f
′ ∈ F , f ′ /∈ Fv.
Because ιv maps Z
Fv into ZF and ιv(R
Fv) ∩ ZF = ιv(Z
Fv), it induces an embedding of the n-dimensional
torus RFv/ZFv into RF/ZF , which we also denote by ιv.
Lemma 3.1 With these notations, RF , ZF , and RF/ZF are the direct sum of n and ιv(RFv), n ∩ Zn and
Fv), and N and ιv(R
Fv/ZFv), respectively.
It follows that N is connected, a torus, with integral lattice equal to n ∩ ZF . It also follows that π ◦ ιv
is an isomorphism from the torus RFv/ZFv onto the torus T .
Proof Let t ∈ RF . Because the Xf , f ∈ Fv, form an R-basis of t, there exists a unique t
v ∈ RFv , such
π(t) =
(tv)f Xf = π(ιv(t
that is, t− ιv(t
v) ∈ n. Moreover, because the Xf , f ∈ Fv , also form a Z-basis of tZ, we have that t
v ∈ ZFv ,
and therefore t− ιv(t
v) ∈ n ∩ ZF , if t ∈ ZF . �
Lemma 3.2 We have z ∈ Z if and only if µ(z) − λ ∈ π∗(t∗). More explicitly, if and only if there exists a
ξ ∈ t∗ such that
|zf |
2/2− λf = 〈Xf , ξ〉 for every f ∈ F. (3.1)
When z ∈ Z , the ξ in (3.1) is uniquely determined.
Furthermore, Z = (µ− λ)−1(π∗(∆)), (µ− λ)(Z) = π∗(∆), and Z is a compact subset of CF .
Proof The kernel of ι∗
is equal to the space of all linear forms on RF which vanish on n := kerπ, and
therefore ker ι∗
is equal to the image of π∗ : t∗ → (RF )∗. Because π is surjective, π∗ is injective, which
proves the uniqueness of ξ.
It follows from (3.1) that 〈Xf , ξ〉 + λf ≥ 0 for every f ∈ F , and therefore ξ ∈ ∆ in view of (2.1).
Conversely, if ξ ∈ ∆, then there exists for every f ∈ F a complex number zf such that |zf |
2/2 = 〈Xf , ξ〉+
λf , which means that z ∈ Z and (µ− λ)(z) = π
∗(ξ). The set π∗(∆) is compact because ∆ is compact and
π∗ is continuous. Because the mapping µ− λ is proper, it follows that Z = (µ− λ)−1(π∗(∆)) is compact.
Let v ∈ V . The Xf , f ∈ Fv , form an R-basis of t, and therefore there exists for each z ∈ C
Fv a unique
ξ = µv(z) ∈ t
∗ such that (3.1) holds for every f ∈ Fv . That is, the mapping µv : C
Fv → t∗ is defined by
the equations
|zf |
2/2 − λf = 〈Xf , µv(z)〉, z ∈ C
Fv , f ∈ Fv. (3.2)
In other words, µv is defined by the formula
ρv ◦ π
∗ ◦ µv = ρv ◦ (µ− λ) ◦ ιv, (3.3)
where ρv denotes the restriction projection from R
F onto RFv .
Lemma 3.3 If we let T act on CFv via RFv/ZFv by means of (t, z) 7→ (π ◦ ιv)−1(t) ·z, then µv : CFv → t∗
is a momentum mapping for this Hamiltonian action of T on CFv , with µv(0) = v. Here the symplectic
form on CFv is equal to
σ := (i /4π)
dzf ∧dzf = (1/2π)
dxf ∧dyf , (3.4)
that is, (2.4) with F replaced by Fv .
Let ρv denote the restriction projection from C
F onto CFv , and let Uv be the interior of the subset ρv(Z)
of CFv . Write
∆v := ∆ \
f ′∈F\Fv
f ′. (3.5)
Then ρv(Z) = µv
−1(∆), µv(ρv(Z)) = ∆, Uv = µv
−1(∆v), and µv(Uv) = ∆v. In particular ρv(Z) is a
compact subset of CFv , and Uv is a bounded and connected open neighborhood of 0 in C
Proof The first statement follows from (3.3), the fact that ρv◦µ◦ιv is a momentum mapping for the standard
Fv/ZFv action on CFv , and the fact that a momentum mapping for a Hamiltonian action plus a constant is a
momentum mapping for the same Hamiltonian action. It follows in view of (3.2) that 〈Xf , µv(0)〉+λf = 0
for every f ∈ Fv , hence µv(0) = v in view of i) in the definition of a Delzant polytope, and the fact that {v}
is the intersection of all the f ∈ Fv .
It follows from (3.2), Lemma 3.2, that z ∈ Z if and only if
|zf |
2/2 = 〈Xf , µv(ρv(z))〉 + λf for every f ∈ F, (3.6)
where we note that these equations are satisfied by definition for the f ∈ Fv. Therefore, if z ∈ Z , then (3.6)
and (2.1) imply that µv(ρ(z)) ∈ ∆. Conversely, if ξ ∈ ∆, then it follows from Lemma 3.2 that there exists
z ∈ Z such that π∗(ξ) = µ(z)− λ, of which the restriction to Fv yields ξ = µv(ρ(z)).
If ξ ∈ ∆v, z
v ∈ CFv , µv(z
v) = ξ, then 〈Xf ′ , µv(z
v)〉 + λf ′ > 0 for every f
′ ∈ F \ Fv, which will
remain valid if we replace zv by z̃v in a sufficiently small neighborhood of zv in CFv . It follows that we can
find z̃ ∈ CF such that ρv(z̃) = z̃
v and (3.6) holds with z replaced by z̃. That is, z̃ ∈ Z , and we have proved
that zv ∈ Uv.
Let conversely z ∈ Uv ⊂ C
Fv . We have in view of (3.2) that
|zf |
2/2 = 〈Xf , µv(z)− µv(0)〉 = 〈Xf , µv(z)− v〉
for every f ∈ Fv. Therefore µv(z) − v is multiplied by c
2 if we replace z by c z, c > 0. Because z is in
the interior of ρv(Z), we have c z
v ∈ ρv(Z), hence µv(c z) ∈ ∆ for c > 1, c sufficiently close to 1. On the
other hand, if ξ belongs to a face of ∆ which is not adjacent to v, then v + τ (ξ − v) /∈ ∆ for any τ > 1. It
follows that µv(z) does not belong to any f
′ ∈ F \ Fv, that is, µv(z) ∈ ∆v. �
The equation (3.6) can be written in the form |zf | = rf (µv(ρv(z))), where, for each f ∈ F , the function
rf : ∆ → R≥0 is defined by
rf (ξ) := (2(〈Xf , ξ〉+ λf ))
1/2, f ∈ F, ξ ∈ ∆. (3.7)
We now view the equations (3.6) for z ∈ Z as equations for the coordinates zf ′ , f
′ ∈ F \ Fv, with the
zf , f ∈ Fv as parameters, where the latter constitute the vector z
v = ρv(z). If z
v ∈ Uv, then for each
f ′ ∈ F \ Fv the coordinate zf ′ lies on the circle about the origin with strictly positive radius rf ′(µv(z
Because Lemma 3.1 implies that the homomorphism which assigns to each element of N its projection to
F\Fv/ZF\Fv is an isomorphism, and the latter torus is the group of the coordinatewise rotations of the zf ′ ,
f ′ ∈ F \ Fv, this leads to the following conclusions.
Proposition 3.4 Let v be a vertex of ∆. The open subset Zv := ρv−1(Uv) ∩ Z of Z is a connected smooth
submanfold of CF of real dimension 2n + (d − n), where d = #(F ) and d − n = dimN . The action
of the torus N on Zv is free, and the projection ρv : Zv → Uv exhibits Zv as a principal N -bundle over
Uv. It follows that we have a reduced phase space Mv := Zv/N , which is a connected smooth symplectic
2n-dimensional manifold, which carries an effective Hamiltonian T -action with momentum mapping as in
(2.7), with Z replaced by Zv.
There is a unique global section sv : Uv → Zv of ρv : Zv → Uv such that sv(z)f ′ ∈ R>0 for every
z ∈ Uv and f
′ ∈ F \ Fv. Actually, sv(z)f ′ = rf ′(µv(z)) when z ∈ Uv and f
′ ∈ F \ Fv , and therefore the
section sv is smooth. If pv : Zv → Mv = Zv/N denotes the canonical projection, then ψv := pv ◦ sv is a
T -equivariant symplectomorphism from Uv onto Mv, where T acts on Uv via R
Fv/ZFv , as in Lemma 3.3.
Remark 3.5 When z belongs to the closure ρv(Z) = Uv of Uv in CFv , see Lemma 3.3, we can define
sv(z) ∈ C
F by sv(z)f = zf when f ∈ Fv and sv(z)f ′ = rf ′(µv(z)) when f
′ ∈ F \ Fv . This defines
a continuous extension sv : Uv → C
F of the mapping sv : Uv → Z . Therefore sv(Uv) ⊂ Z , and
ψv := p ◦ sv : Uv →M is a continuous extension of the diffeomorphism ψv : Uv →Mv .
The continuous mapping ψv : Uv → M is surjective, but the restriction of it to the boundary ∂Uv :=
Uv \Uv of Uv in C
Fv is not injective. If zv ∈ ∂Uv, then the set G of all f
′ ∈ F \ Fv such that sv(z
v)f ′ = 0,
or equivalently µT (ψv(z
v)) ∈ f ′, is not empty. The fiber of ψv over ψv(z
v) is equal to the set of all tv · zv,
where the tv ∈ RFv/ZFv are of the form
tvf = −
(v)fg tg, f ∈ Fv,
where tg ∈ R/Z. It follows that each fiber is an orbit of some subtorus of R
Fv/ZFv acting on CFv . ⊘
Recall the definition (3.5) of the open subset ∆v of the Delzant polytope ∆. Because the union over all
vertices v of the ∆v is equal to ∆, we have the following corollary.
Corollary 3.6 The sets Zv, v ∈ V , form a covering of Z . As a consequence, Z is a smooth submanifold
of CF of real dimension n + d. The action of the torus N on Z is free, and we have a reduced phase
space M := Z/N , which is a compact and connected smooth 2n-dimensional symplectic manifold, which
carries an effective Hamiltonian T -action with momentum mapping µT :M → T as in (2.7). The sets Mv,
v ∈ V , form an open covering of M and the ϕv := (ψv)
−1 : Mv → Uv form an atlas of T -equivariant
symplectic coordinatizations of the Hamiltonian T -space M . For each v ∈ V , we have Mv = µT
−1(∆v),
and µT |Mv = µv ◦ ϕv.
For a characterization of Mv in terms of the orbit type stratification in M for the T -action, see Corollary
5.5, which also implies that Mv is an open cell in M .
Corollary 3.7 For every f ∈ F the set µT−1(f) is a real codimension two smooth compact connected
smooth symplectic submanifold of M .
For each v ∈ V , the set Mv is dense in M , and the diffeomorphism ψv : Uv → Mv is maximal among
all diffeomorphisms from open subsets of CFv onto open subsets of M .
Proof If f ∈ F , then for each v ∈ V we have that
−1(f) = {z ∈ Uv | zf = 0} (3.8)
if v ∈ f , that is, f ∈ ∆v. This follows from (3.2) and i) in the description of ∆ in the beginning of Section
2. On the other hand, µv
−1(f) = ∅ if f /∈ ∆v. Because µT
−1(f) ∩Mv = ψv(µv
−1(f)), and the Mv,
v ∈ V , form an open covering of M , this proves the first statement. The second statement follows from the
first one, because the complement of Mv in M is equal to the union of the sets µT
−1(f ′) with f ′ ∈ F \ Fv.
Remark 3.8 It follows from the proof of Corollary 3.7, that µT−1(f) is a connected component of the
fixed point set in M of the of the circle subgroup exp(RXf ) of T .
Actually, µT
−1(f) is a Delzant space for the action of the (n− 1)-dimensional torus
T/exp(RXf ),
with Delzant polytope P ⊂ (t/(RXf ))
∗ such that the image of P in t∗ under the embedding (t/(RXf ))
∗ is equal to a translate of f .
In a similar way, if g is a k-dimensional face of ∆, then µT
−1(g) is a 2k-dimensional Delzant space for
the quotient of T by the subtorus of T which acts trivially on µT
−1(g). ⊘
Remark 3.9 Let, for each f ∈ F , cf ∈H
2(M, Z) ⊂H2(M, R) denote the Poincaré dual of the codimen-
sion two Delzant subspace µT
−1(f) of M , see Remark 3.8. Then, with our normalization of the symplectic
form (2.4), the de Rham cohomology class [σM ] of the symplectic form σM of the Delzant space is equal to
[σM ] =
λf cf , (3.9)
see Guillemin [8, Thm. 6.3]. In particular [σM ] ∈H
2(M, Z), and therefore [σM ] is equal to the Chern class
of a complex line bundle over M , if all the coefficients λf , f ∈ F , are integers.
If ∆ is a simplex, when M is isomorphic to the n-dimensional complex projective space, then the
−1(f), f ∈ F , are complex projective hyperplanes, see Subsection 6.1, which are all homologous to
each other. It follows that in this case [σM ] = γ c, where c is the Poincaré dual of a complex projective
hyperplane and γ is equal to the sum of all the coefficients λf , f ∈ F . ⊘
Remark 3.10 Let ι : T → Rn/Zn be an isomorphism of tori, which allows us to let t ∈ T act on Cn via
n/Zn by means of
(t · z)j = e
2π i ι(t)j zj, 1 ≤ j ≤ n.
Let U be a connected T -invariant open neighborhood of 0 in Cn, provided with the symplectic form (2.4)
with F replaced by {1, . . . , n}. Let ψ : U → M be a T -equivariant symplectomorphism from U onto an
open subset ψ(U) of M . Because 0 is the unique fixed point for the T -action in U , and the fixed points
for the T -action in M are the pre-images under µT of the vertices of ∆, there is a unique v ∈ V such
that µT (ψ(0)) = v. Let Iv : C
Fv → Cn denote the complex linear extension of the tangent map of the
torus isomorphism ι ◦ (π ◦ ιv). In terms of the notation of Lemma 3.3 and Proposition 3.4, we have that
U ⊂ Iv(Uv) and ψv = ψ ◦ Iv on Iv
−1(U), which leads to an identification of ψ with the restriction of ψv to
the connected open subset Iv
−1(U) of Uv, via the isomorphism Iv
The ψ’s, with U equal to a ball in Cn centered at the origin, are the equivariant symplectic ball embed-
dings in Pelayo [10], and the second statement in Corollary 3.7 shows that the diffeomorphisms ψv are the
maximal extensions of these equivariant symplectic ball embeddings. ⊘
4 The coordinate transformations
Let v, w ∈ V . Then
Uv, w := ϕv(Mv ∩Mw) = Uv ∩ ψv
−1 ◦ ψw(Uw)
= {zv ∈ Uv | (z
v)f 6= 0 for every f ∈ Fv \ Fw}. (4.1)
In this section we will give an explicit formula for the coordinate transformations
ϕw ◦ ϕv
−1 = ψw
−1 ◦ ψv : Uv, w → Uw, v,
which then leads to a description of the Delzant space M as obtained by gluing together the subsets Uv with
the coordinate transformations as the gluing maps.
Let f ∈ F . Because the Xg , g ∈ Fw, form a Z-basis of tZ, and Xf ∈ tZ, there exist unique integers
f , g ∈ Fw, such that
f Xg. (4.2)
Note that if f ∈ Fw, then (w)
f = 1 when g = f and (w)
f = 0 otherwise. For the following lemma recall
that rg is defined by expression (3.7).
Lemma 4.1 Let v, w ∈ V , zv ∈ Uv, w. Then zw := ϕw ◦ ϕv−1(zv) ∈ Uw ⊂ CFw is given by
zwg =
(zvf )
f∈Fv\Fw
|zvf |
f (4.3)
if g ∈ Fw ∩ Fv, and
zwg =
(zvf )
f rg(µv(z
f∈Fv\Fw
|zvf |
f (4.4)
if g ∈ Fw \ Fv.
Proof The element zw ∈ Uw is determined by the condition that sw(zw) belongs to the N -orbit of sv(zv).
That is,
w)f = e
i tf sv(z
v)f for every f ∈ F
for some t ∈ RF such that ∑
tf Xf = 0. (4.5)
It follows from (4.5), (4.2) and the linear independence of the Xg, g ∈ Fw, that t ∈ n if and only if
tg = −
f∈F\Fw
tf for every g ∈ Fw. (4.6)
Note that µv(z
v) = µT (m) = µw(z
w), where m = ψv(z
v) = ψw(z
w). It follows from the definition of
the sections sv and sw, see Proposition 3.4, that
i) sv(z
v)f = z
f and sw(z
w)f = z
f if f ∈ Fv ∩ Fw,
ii) sv(z
v)f = z
f and sw(z
w)f = rf (µw(z
w)) = rf (µv(z
v)) if f ∈ Fv \ Fw,
iii) sv(z
v)f = rf (µv(z
v)) and sw(z
w)f = z
f if f ∈ Fw \ Fv, and
iv) sv(z
v)f = rf (µv(z
v)) = rf (µw(z
w)) = sw(z
w)f if f ∈ F \ (Fv ∪ Fw).
It follows from ii) and iv) that tf = −arg z
f and tf = 0 modulo 2π if f ∈ Fv \Fw and f ∈ F \ (Fv ∪Fw),
respectively. Then (4.6) implies that, modulo 2π,
f∈Fv\Fw
f arg z
f for every g ∈ Fw.
It now follows from i) and iii) that if g ∈ Fw, then z
g = sw(z
w)g = e
i tg sv(z
v)g is equal to
ei tg zvg =
(zvf )
f∈Fv\Fw
|zvf |
if g ∈ Fv, and equal to
ei tg |zvg | =
(zvf )
f |zvg |/
f∈Fv\Fw
|zvf |
if g /∈ Fv , respectively. Here we have used that if g ∈ Fw, then (w)
= 1 if f = g and (w)g
= 0 if f ∈ Fw,
f 6= g. Because |zvg | = rg(µv(z
v)) if g /∈ Fv, see (3.6) and (3.7), this completes the proof of the lemma. �
Remark 4.2 Note that zv ∈ Uv, w means that zv ∈ Uv and zvf 6= 0 if f ∈ Fv \ Fw. Furthermore, z
v ∈ Uv
implies that if g /∈ Fv , then µv(z
v) /∈ g, and therefore rg is smooth on a neighborhood of µv(z
v). Finally,
note that if g ∈ Fw and f ∈ Fv ∩Fw, then (w)
∈ {0, 1}, and therefore each of the factors in the right hand
sides of (4.3) and (4.4) is smooth on Uv, w. ⊘
Remark 4.3 In (4.3) and (4.4) only the integers (w)gf appear with f ∈ Fv and g ∈ Fw. Let (w v) denote the
matrix (w)gf , where f ∈ Fv and g ∈ Fw. Then (w v) is invertible, with inverse equal to the integral matrix
(v w). These integral matrices also satisfy the cocycle condition that (w v) (v u) = (w u), if u, v, w ∈ V .
These properties follow from the fact that (4.2) shows that (w v) is the matrix which maps the Z-basis Xg,
g ∈ Fw, onto the Z-basis Xf , f ∈ Fv, of tZ. It is no surprise that these base changes enter in the formulas
which relate the models in the vector spaces CFv for the different choices of v ∈ V . ⊘
Corollary 4.4 Let, for each v ∈ V , the mapping µv : CFv → t∗ be defined by (3.2), which is a momentum
mapping for a Hamiltonian T -action via RFv/ZFv on the symplectic vector space CFv as in Lemma 3.3.
Define Uv := µv
−1(∆v). If also w ∈ V , define Uv, w as the right hand side of (4.1), and, if z
v ∈ Uv, w,
define ϕw, v(z
v) := zw, where zw ∈ CFw is given by (4.3) and (4.4).
Then ϕw, v is a T -equivariant symplectomorphism from Uv, w onto Uw, v such that µw = µv ◦ ϕv, w
on Uw, v. The ϕw, v satisfy the cocycle condition ϕw, v ◦ ϕv, u = ϕw, u where the left hand side is defined.
Glueing together the Hamiltonian T -spaces Uv, v ∈ V , with the momentum maps µv, by means of the
gluing maps ϕw, v, v, w ∈ V , we obtain a compact connected smooth symplectic manifold M̃ with an
effective Hamiltonian T -action with a common momentum map µ̃ : M̃ → T such that µ̃(M̃ ) = ∆. In other
words, M̃ is a Delzant space for the Delzant polytope ∆.
The Delzant space M̃ is obviously isomorphic to the Delzant spaceM = µ−1({λ})/N introduced in Section
2, and actually the isomorphism is used in the proof that M̃ is a Delzant space for the Delzant polytope ∆.
The only purpose of Corollary 4.4 is to exhibit the Delzant space as obtained from gluing together the Uv,
v ∈ V , by means of the gluing maps ϕv, w, v, w ∈ V .
5 The toric variety
Let T := {z ∈ C | |z| = 1} denote the unit circle in the complex plane. The mapping t 7→ u where
uf = e
2π i tf for every f ∈ F is an isomorphism from the torus RF/ZF onto TF , where TF acts on CF by
means of coordinatewise multiplication and RF/ZF acted on CF via the isomorphism from RF/ZF onto
F . The complexification TC of the compact Lie group T is the multiplicative group C
× of all nonzero
complex numbers, and the complexification of TF is equal to TF
:= (TC)
F = (C×)F , which also acts on
F by means of coordinatewise multiplication.
The complexification NC of N is the subgroup exp(nC) of U
, where nC := n⊕ i n ⊂ C
F denotes the
complexification of n, viewed as a complex linear subspace, a complex Lie subalgebra, of the Lie algebra
F of TF
. In view of (4.6), we have, for every v ∈ V , that NC is equal to the set of all t ∈ T
such that
f∈F\Fv
f , g ∈ Fv. (5.1)
This implies that NC is a closed subgroup of T
isomorphic to T
, and therefore NC is a reductive
complex algebraic group.
If we define
v := {z ∈ C
F | zf 6= 0 for every f ∈ F \ Fv}, (5.2)
then it follows from (5.1) that the action of NC on C
v is free and proper. It follows that the action of NC on
v (5.3)
is free and proper, and therefore the NC-orbit space
Mtoric := CF∆/NC (5.4)
has a unique structure of a complex analytic manifold of complex dimension n such that the canonical pro-
jection from CF∆ onto M
toric exhibits CF∆ as a principal NC-bundle over M
toric. On Mtoric we still have the
complex analytic action of the complex Lie group group TF
/NC, which is isomorphic to the complexifi-
cation TC of our real torus T induced by the projection π. The complex analytic manifold M
toric together
with the complex analytic action of TC on it is the toric variety defined by the polytope ∆ in the title of this
section.
If v ∈ V and z ∈ CFv , then it follows from (5.1) that there is a unique t ∈ NC such that tf = zf for
every f ∈ F \ Fv, or in other words, z = t · ζ , where ζ ∈ C
F is such that ζf = 1 for every f ∈ F \ Fv.
Let Sv : C
Fv → CFv be defined by Sv(z
v)f = z
f when f ∈ F and Sv(z
v) = 1 when f ∈ F \ Fv, as in
Audin [2, p. 159]. If Pv : C
v → C
v /NC denotes the canonical projection from C
v onto the open subset
Mtoricv := C
v /NC of M
toric, then Ψv := Pv ◦ Sv is a complex analytic diffeomorphism from C
Fv onto
Mtoricv . It is TC-equivariant if we let TC act on C
Fv via TFv
as in Lemma 3.3. We use the diffeomorphism
Φv := Ψv
−1 from Mtoricv onto C
Fv as a coordinatization of the open subset Mtoricv of M
toric.
If v, w ∈ V , then
Utoricv, w := Φv(M
toric
toric
w ) = C
Fv ∩Ψv
−1 ◦Ψw(C
= {zv ∈ CFv | (zv)f 6= 0 for every f ∈ Fv \ Fw}. (5.5)
Moreover, with a similar argument as for Lemma 4.1, actually much simpler, we have that for every zv ∈
Utoricv, w the element z
w := Φw ◦Φv
−1(zv) ∈ CFw is given by
zwg =
(zvf )
f , g ∈ Fw. (5.6)
In this way the coordinate transformation Φw◦Φv
−1 is a Laurent monomial mapping, much simpler than the
coordinate transformation (4.3), (4.4). It follows that the toric variety Mtoric can be alternatively described
as obtained by gluing the n-dimensional complex vector spaces CFv , v ∈ V , together, with the maps (5.6)
as the gluing maps. This is the kind of toric varieties as introduced by Demazure [4, Sec. 4].
For later use we mention the following observation of Danilov [5, Th. 9.1], which is also of interest in
itself.
Lemma 5.1 Mtoric is simply connected.
Proof Let w ∈ V . It follows from (5.5), for all v ∈ V , that the complement of Mtoricw in Mtoric is equal to
the union of finitely closed complex analytic submanifolds of complex codimension one, whereas Mtoricw is
contractible because it is diffeomorphic to the complex vector space CFw . Because complex codimension
one is real codimension two, any loop in Mtoric with base point in Mtoricw can be slightly deformed to such
a loop which avoids the complement of Mtoricw in Mtoric, that is, which is contained in M
toric
w , after which it
can be contracted within Mtoricw to the base point in M
toric
w . �
Recall the definition in Section 2 of the reduced phase space M = Z/N .
Theorem 5.2 The identity mapping from Z into CF∆, followed by the canonical projection P from C
Mtoric = CF∆/NC, induces a T -equivariant diffeomorphism ̟ from M = Z/N onto M
toric. It follows that
each NC-orbit in C
∆ intersects Z in an N -orbit in Z .
Proof Because N is a closed Lie subgroup of NC, we have that the mapping P : Z → CF∆/NC induces a
mapping ̟ : Z/N → CF∆/NC, which moreover is smooth.
If v ∈ V , z ∈ Zv, then it follows from (5.1) that the tf , f ∈ F \ Fv , of an element t ∈ NC can take
arbitrary values, and therefore the |zf |, f ∈ F \ Fv can be moved arbitrarily by means of infinitesimal
NC-actions. Because Z is defined by prescribing the |zf |, f ∈ F \ Fv, as a smooth function of the zf ,
f ∈ Fv, and the Zv, v ∈ V , form an open covering of Z , this shows that at each point of Z the NC-orbit is
transversal to Z , which implies that ̟ is a submersion.
It follows that ̟(M) is an open subset of Mtoric. Because M is compact and ̟ is continuous, ̟(M)
is compact, and therefore a closed subset of Mtoric. Because Mtoric is connected, the conclusion is that
̟(M) =Mtoric, that is, ̟ is surjective.
Because ̟ is a surjective submersion, dimRM = 2n =dimRM
toric, and M is connected, we conclude
that ̟ is a covering map. Because Mtoric is simply connected, see Lemma 5.1, we conclude that ̟ is
injective, that is, ̟ is a diffeomorphism. �
Remark 5.3 Theorem 5.2 is the last statement in Delzant [3], with no further details of the proof. Audin
[2, Prop. 3.1.1] gave a proof using gradient flows, whereas the injectivity has been proved in [7, Sec. A1.2]
using the principle that the gradient of a strictly convex function defines an injective mapping. ⊘
Note that in the definition of the toric variety Mtoric, the real numbers λf , f ∈ F , did not enter, whereas
these numbers certainly enter in the definition of M , the symplectic form on M , and the diffeomorphism
̟ :M →Mtoric. Therefore the symplectic form σtoricλ := (̟
−1)∗(σ) on Mtoric on Mtoric will depend on the
choice of λ ∈ RF . On the symplectic manifold (Mtoric, σtoricλ ), the action of the maximal compact subgroup
T of TC is Hamiltonian, with momentum mapping equal to
µtoricλ := µ ◦̟
−1 :Mtoric → t∗, (5.7)
where µtoricλ (M
toric) = ∆, where we note that ∆ in (2.1) depends on λ.
In the following lemma we compare the reduced phase space coordinatizations with the toric variety
coordinatizations.
Lemma 5.4 Let v ∈ V . Then Mtoricv = ̟(Mv), and
θv := Ψv
−1 ◦̟ ◦ ψv (5.8)
is a TFv-equivariant diffeomorphism from Uv onto C
For each zv ∈ Uv, the element ζ
v := θv(z
v) is given in terms of zv by
ζvf = z
f ′∈F\Fv
rf ′(µv(z
f ′ , f ∈ Fv , (5.9)
where the functions rf ′ : ∆ → R≥0 are given by (3.7). We have
µv(zv) = µT (ψv(z
v)) = µtoricλ (Ψv(ζ
v)), (5.10)
and zv = θv
−1(ζv) is given in terms of ζ
zvf = ζ
f ′∈F\Fv
rf ′(ξ)
f ′ , f ∈ Fv, (5.11)
where ξ is the element of ∆ equal to the right hand side of (5.10).
Proof It follows from Lemma 3.3 and the paragraph preceding Proposition 3.4 that if zv ∈ ρv(Z), then
zv ∈ Uv if and only if z
f ′ 6= 0 for every f
′ ∈ F \ Fv. That is, the set Zv in Proposition 3.4 is equal to
Z ∩ CFv . It therefore follows from Theorem 5.2 that each NC-orbit in the NC-invariant subset C
v of C
intersects the N -invariant subset Zv of Z in an N -orbit in Zv, that is,
Mtoricv = Pv(C
v ) = ̟(pv(Zv)) = ̟(Mv).
If zv ∈ Uv, then Proposition 3.4 implies that sv(z
v)f = z
f for every f ∈ Fv and
v)f ′ = rf ′(µv(z
v)), f ′ ∈ F \ Fv.
If we define t ∈ TF
tf ′ = rf ′(µv(z
v))−1, f ′ ∈ F \ Fv,
f ′∈F\Fv
rf ′(µv(z
f ′ , f ∈ Fv ,
then (t · sv(z
v))t′ = 1 for every t
′ ∈ F \ Fv and, for every f ∈ Fv, ζ
f := (t · sv)f is equal to the right hand
side of (5.9). That is, t · sv(z
v) = Sv(ζ
v), see the definition of Sv in the paragraph preceding (5.5). On the
other hand, it follows from (5.1) that t ∈ NC, and therefore
Ψv(ζv) = Pv(t · sv(z
v)) = Pv(sv(z
v)) = ̟ ◦ pv(sv(z
v)) = ̟ ◦ ψv(z
that is, ζv = Ψv
−1 ◦̟ ◦ ψv(z
v). �
Corollary 5.5 Let s be the relative interior of a face of ∆. Then µT−1(s) is equal to a stratum S of the
orbit type stratification in M of the T -action, and also equal to the preimage under ̟ : M → Mtoric of a
TC-orbit in M
toric. If s = {v} for a vertex v, then µT
−1(s) = {mv} for the unique fixed point mv in M for
the T -action such that µT (mv) = v.
The mapping s 7→ µT
−1(s) is a bijection from the set Σ∆ of all relative interiors of faces of ∆ onto the
set Σ of all strata of the orbit type stratificiation in M for the action of T . If s, s′ ∈ Σ∆ then s is contained
in the closure of s′ in ∆ if and only if µT
−1(s) is contained in the closure of µT
−1(s′) in M .
The domain of definition Mv of ϕv in M is equal to the union of the S ∈ Σ such that mv belongs
to the closure of S in M . The domain of definition Mtoricv = ̟(Mv) of Φv is equal to the union of the
corresponding strata of the T -action in Mtoric, each of which is a TC-orbit in M
toric. Mv and M
toric
v are
open cells in M and Mtoric, respectively.
Proof There exists a vertex v of ∆ such that v belongs to the closure of s in t∗, which implies that s is
disjoint from all f ′ ∈ F \ Fv. Let Fv, s denote the set of all f ∈ Fv such that s ⊂ f , where Fv, s = ∅ if and
only if s is the interior of ∆. For any subset G of Fv, let C
G denote the set of all z ∈ C
Fv such that zf = 0
if f ∈ G and zf 6= 0 if f ∈ Fv \G. It follows from µv = µT ◦ ψv and (3.8) that ψ
v (µT
−1(s)) is equal to
Uv ∩ C
G with G = Fv, s. The diffeomorphism θv maps this set onto the set C
G with G = Fv, s. Because
the sets of the form CFvG with G ⊂ Fv are the strata of the orbit type stratification of the T
Fv -action on CFv ,
and also equal to the (TC)
Fv -orbits in CFv , the first statement of the corollary follows.
The second statement follows from µv
−1({v}) = {0} and the fact that 0 is the unique fixed point of the
Fv-action in Uv.
If s ∈ Σ∆ and v ∈ V , then mv belongs to the closure of µT
−1(s) if and only if s is not contained in any
f ′ ∈ F \ Fv. This proves the characterization of the domain of definition Mv := Zv/N = µT
−1(∆v) of
ϕv. The last statement follows from the fact that Φv is a diffeomorphism from M
toric
v onto the vector space
Fv , and ̟ is a diffeomorphism from Mv onto M
toric
v . �
Remark 5.6 If v, w ∈ V , then
ϕw ◦ ϕv
−1 = ψw
−1 ◦ ψv = θw
−1 ◦ (Ψw
−1 ◦Ψv) ◦ θv = θw
−1 ◦ (Φw ◦Φv
−1) ◦ θv.
Using the formula (5.6) for Φw ◦ Φv
−1, this can be used in order to obtain the formulas (4.3), (4.4) as a
consequence of (5.9). In the proof, it is used that ξ := µv(z
v) = µw(z
w), |zvf | = rf (ξ) if f ∈ Fv \ Fw, and
= (w)
if f ′ ∈ F \ Fv and g ∈ Fw. ⊘
In the following corollary we describe the symplectic form σtoricλ on the toric variety M
toric in the toric
variety coordinates.
Corollary 5.7 For each v ∈ V , the symplectic form (Φv
−1)∗(σtoricλ ) on C
Fv is equal to (θv
−1)∗(σv), where
σv is the standard symplectic form on C
Fv given by (3.4).
Because rf ′(µv(z
v))2 is an inhomogeneous linear function of the quantities |zvf |
2, it follows from (5.9) that
the equations which determine the |zvf |
2 in terms of the quantities |ζvf |
2 are n polyomial equations for the n
unknowns |zfv |2, f ∈ Fv , where the coefficients of the polyomials are inhomogeneous linear functions of the
|ζvf |, f ∈ Fv. In this sense the |z
2, f ∈ Fv, are algebraic functions of the |ζ
2, f ∈ Fv, and substituting
these in (5.9) we obtain that the diffeomorphism θv
−1 from CFv onto Uv is an algebraic mapping. If ∆ is a
simplex, when Mtoric is the n-dimensional complex projective space, we have an explicit formula for θv
see Subsection 6.1. However, already in the case that ∆ is a planar quadrangle, when Mtoric is a complex
two-dimensional Hirzebruch surface, we do not have an explicit formula for θv
−1. See Subsection 6.2.
Summarizing, we can say that in the toric variety coordinates the complex structure is the standard
one and the coordinate transformations are the relatively simple Laurent monomial transformations (5.6).
However, in the toric variety coordinates the λ-dependent symplectic form in general is given by quite
complicated algebraic functions. On the other hand, in the reduced phase space coordinates the symplectic
form is the standard one, but the coordinate transformations (4.3), (4.4) are more complicated. Also the
complex structure in the reduced phase space coordinates, which depends on λ, is given by more complicated
formulas.
Remark 5.8 It is a challenge to compare the formula in Corollary 5.7 for the symplectic form in toric
variety coordinates with Guillemin’s formula in [7, Th. 3.5 on p. 141] and [8, (1.3)]. Note that in the
latter the pullback by means of the momentum mapping appears of a function on the interior of ∆, where in
general we do not have a really explicit formula for the momentum mapping in toric variety coordinates. ⊘
6 Examples
6.1 The complex projective space
Let ∆ be an n-dimensional simplex in t∗. A little bit of puzzling shows that there is a Z-basis ei, 1 ≤ i ≤ n,
of the integral lattice tZ in t, such that, with the notation
e0 = −
ei, (6.1)
the Xf , f ∈ F , are the ei, 0 ≤ i ≤ n. That is, in the sequel we write F = {0, 1, . . . , n}. The Delzant
simplex (2.1) is determined by the inequalities 〈ei, ξ〉+ λi ≥ 0, 0 ≤ i ≤ n, which has a non-empty interior
if and only if
λi > 0. (6.2)
In the sequel we take for v the vertex determined by the equations 〈ei, ξ〉+ λi = 0 for all 1 ≤ i ≤ n, where
Fv = {1, . . . , n}. If we write ξi = 〈ei, ξ〉, 1 ≤ i ≤ n, when ξ ∈ t
∗, then (3.2) yields that
v)i = |zi|
2/2− λi, 1 ≤ i ≤ n.
It follows from (3.7) that
r0(ξ) = (2(−
ξi + λ0))
and therefore (5.9) yields that
ζvi = z
i (2γ − ‖z
v‖2)−1/2, 1 ≤ i ≤ n, (6.3)
where we have written
‖zv‖2 =
|zvi |
Note that Uv is the open ball in C
n with center at the origin and radius equal to (2c)1/2.
The equations (6.3) imply that
‖ζv‖2 = ‖zv‖
2/(2γ − ‖zv‖2),
hence
‖zv‖2 = 2γ ‖ζv‖2/(1 + ‖ζv‖2).
Therefore the mapping θv−1 : ζv 7→ zv is given by the explicit formulas
zvi = ζ
i (2γ/(1 + ‖ζ
v‖2))1/2, 1 ≤ i ≤ n. (6.4)
It can be verified that the symplectic form (θv
−1)∗(σv), where
σv = (1/2π)
dxvi ∧dy
is the standard symplectic form in (3.4), is equal to γ times the Fubini-Study form in Griffiths and Harris
[6, p. 30, 31]. In view of Remark 3.9 this agrees with the fact that the de Rham cohomology class of the
Fubini-Study form is Poincaré dual to the homology class of a complex projective hyperplane in the complex
projective space, see Griffiths and Harris [6, p. 122].
6.2 The Hirzebruch surface
Let n = 2 and let ∆ be a quadrangle in the t∗ plane. A little bit of puzzling shows that there is an m ∈ Z≥0
and a Z-basis e1, e2 of the integral lattice tZ in t, such that the Xf , f ∈ F , are the ei, 1 ≤ i ≤ 4, with
e3 = −e1 +me2, and e4 = −e2. We recognize the toric variety M
toric as the Hirzebruch surface Σm, see
Hirzebruch [9].
The Delzant polytope (2.1) is determined by the inequalities 〈ei, ξ〉 + λi ≥ 0, 1 ≤ i ≤ 4, which is a
quadrangle if and only if
γ± := λ1 + λ3 ±mλ4 > 0, (6.5)
which inequalities imply that λ2 + λ4 = γ+ + γ− > 0.
In the sequel we take for v the vertex determined by the equations 〈ei, ξ〉+ λi = 0 for i = 1, 2, where
Fv = {1, 2}. If we write ξi = 〈ei, ξ〉, 1 ≤ i ≤ 2, when ξ ∈ t
∗, then (3.2) yields that
v)i = |zi|
2/2 − λi, 1 ≤ i ≤ 2.
It follows from (3.7) that
r3(ξ) = (2(−ξ1 +mξ2 + λ3))
1/2, r4(ξ) = (2(−ξ2 + λ4))
and therefore (5.9) yields that
ζv1 = z
1 (2γ−|z
2 +m |zv2 |
2)−1/2, (6.6)
ζv2 = z
2 (2γ−|z
2 +m |zv2 |
2)m/2 (2(γ+ + γ−)− |z
2)−1/2. (6.7)
If we write ti = |z
2 and τi = |ζ
2, then this leads to the equations
τ1 = t1/(2γ− − t1 +mt2),
τ2 = t2 (2γ− + t1 +mt2)
m/(2(γ+ + γ−)− t2)
for t1, t2. If we solve t1 from the first equation,
t1 = (2γ− +mt2) τ1/(1 + τ1),
and substitute this into the second equation, then this leads to the polynomial equation
(1 + τ1)
m τ2 (2(γ+ + γ−)− t2) = t2 (2γ− +mt2)
m (6.8)
of degree m + 1 for t2. If we substract the left hand side from the right hand side then the derivative with
respect to t2 is strictly positive, and one readily obtains that for every τ1, τ2 ∈ R≥0 there is a unique solution
t2 ∈ R≥0, confirming the first statement in Lemma 5.4.
On the other hand, if we work over C, and view both the parameter ε := (1 + τ1)
m τ2 and the unknown
t2 as elements of the complex projective line P
1, then the equation (6.8) defines a complex algebraic curve
C in the (t2, ε)-plane P
1 × P1, where the restriction to C of the projection to the first variable t2 is a
complex analytic diffeomorphism from C onto P1, as on C we have that ε is a complex analytic function
of t2. In particular C is irreducible. The restriction to C of the projection to the second variable ε is an
(m+ 1)-fold branched covering. Over ε = 0 and over ε = ∞ we have that m of the m+ 1 branches come
together, whereas there are two more branch points on the ε-line over which only two of the branches come
together. The fact that C is irreducible implies that the part of C over the complement of the branch points is
connected, and therefore the analytic continuation of any solution t2 of (6.8), as a complex analytic function
of ε in the complement of the branch points, will reach each other branch if ε runs over a suitable loop. In
other words, the solution t2 is an algebraic function of ε of degree m+ 1, and no branch of a solution is of
lower degree. This holds in particular for our solutions t2 ∈ R≥0 for ε ∈ R≥0.
References
[1] R. Abraham and J.E. Marsden: Foundations of Mechanics. Benjamin/Cummings Publ. Co., London,
etc., 1978.
[2] M. Audin: The Topology of Torus Actions on Symplectic Manifolds. Birkhäuser, Basel, Boston, Berlin,
1991.
[3] T. Delzant: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math.
France 116 (1988) 315–339.
[4] M. Demazure: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. scient. Éc.
Norm. Sup. 3 (1970) 507–588.
[5] V.I. Danilov: The geometry of toric varieties. Russ. Math. Surveys 33:2 (1978) 97–154, translated from
Uspekhi Mat. Nauk SSSR 33:2 (1978) 85–134.
[6] P. Griffiths and J. Harris: Principles of Algebraic Geometry. J. Wiley & Sons, Inc., New York, etc.,
1978.
[7] V. Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces. Birkhäuser,
Boston, etc., 1994.
[8] V. Guillemin: Kaehler structures on toric varieties. J. Differential Geometry 40 (1994) 285–309.
[9] F. Hirzebruch: Über eine Klasse von einfach-zusammenhn̈genden komplexen Mannigfaltigkeiten.
Mathematische Annalen 124 (1951) 77–86.
[10] A. Pelayo: Topology of spaces of equivariant symplectic embeddings. Proc. Amer. Math. Soc. 135
(2007) 277–288.
[11] R.T. Rockafellar: Convex Analysis. Princeton University Press, princeton, N.J., 1970.
J.J. Duistermaat
Mathematisch Instituut, Universiteit Utrecht
P.O. Box 80 010, 3508 TA Utrecht, The Netherlands
e-mail: duis@math.uu.nl
A. Pelayo
Department of Mathematics, University of Michigan
2074 East Hall, 530 Church Street, Ann Arbor, MI 48109–1043, USA
e-mail: apelayo@umich.edu
	Introduction
	The reduced phase space
	The reduced phase space coordinatizations.
	The coordinate transformations
	The toric variety
	Examples
	The complex projective space
	The Hirzebruch surface
