Reduced phase space and toric variety coordinatizations of Delzant spaces

TL;DR

This paper describes explicit coordinate systems for Delzant spaces from symplectic and algebraic geometry perspectives, providing formulas and proofs that clarify their structure and relations.

Contribution

It introduces explicit coordinate transformations for Delzant spaces as reduced phase spaces and toric varieties, enhancing understanding of their geometric structures.

Findings

Explicit formulas for coordinate transformations between reduced phase space and toric variety

Identification of maximal coordinate neighborhoods around fixed points

Simplified proofs of fundamental properties of Delzant spaces

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.

Equations

$$F_v=\{f\in F\mid v\in f\}.$$

$$\Delta =\{\xi \in {\mathfrak{t}}^{\ast }\mid ⟨X_f,\xi ⟩+{\lambda }_f\ge 0\text{mbox}\mathrm{forevery}f\in F\}.$$

$$\pi (t):=\sum _{f\in F}{t}_f{X}_f,t\in {\mathbb{R}}^F.$$

$$(t\cdot z{)}_f=e^{2\pi \text{mbox}\mathrm{i}{t}_f}{z}_f,f\in F.$$

$$(Y\cdot z{)}_f=2\pi i{Y}_f{z}_f,$$

$$z\mapsto ⟨Y,\mu (z)⟩=\sum _{f\in F}{Y}_f|z_f{|}^2/2=\sum _{f\in F}{Y}_f({x_f}^2+{y_f}^2)/2,$$

$$\sigma :=(i/4\pi )\sum _{f\in F}d{z}_f\wedge d{\overline{z}}_f=(1/2\pi )\sum _{f\in F}d{x}_f\wedge d{y}_f,$$

$$\mu (z{)}_f=|z_f{|}^2/2=({x_f}^2+{y_f}^2)/2,f\in F.$$

$${\mu }_N:={\iota }_{\mathfrak{n}}^{\ast }\circ \mu :{\mathbb{C}}^F\to {\mathfrak{n}}^{\ast },$$

$${\pi }^{\ast }\circ {\mu }_T\circ p=(\mu -\lambda ){|}_Z.$$

$$\pi (t)=\sum _{f\in F_v}(t^v{)}_f{X}_f=\pi ({\iota }_v(t^v)),$$

$$|z_f{|}^2/2-{\lambda }_f=⟨X_f,\xi ⟩\text{mbox}\mathrm{forevery}f\in F.$$

$$|z_f{|}^2/2-{\lambda }_f=⟨X_f,{\mu }_v(z)⟩,z\in {\mathbb{C}}^{F_v},f\in F_v.$$

$${\rho }_v\circ {\pi }^{\ast }\circ {\mu }_v={\rho }_v\circ (\mu -\lambda )\circ {\iota }_v,$$

$$\sigma :=(i/4\pi )\sum _{f\in F_v}d{z}_f\wedge d{\overline{z}}_f=(1/2\pi )\sum _{f\in F_v}d{x}_f\wedge d{y}_f,$$

$${\Delta }_v:=\Delta \setminus \bigcup _{f^{\prime }\in F\setminus F_v}{f}^{\prime }.$$

$$|z_f{|}^2/2=⟨X_f,{\mu }_v({\rho }_v(z))⟩+{\lambda }_f\text{mbox}\mathrm{forevery}f\in F,$$

$$|z_f{|}^2/2=⟨X_f,{\mu }_v(z)-{\mu }_v(0)⟩=⟨X_f,{\mu }_v(z)-v⟩$$

$$r_f(\xi ):=(2(⟨X_f,\xi ⟩+{\lambda }_f){)}^{1/2},f\in F,\xi \in \Delta .$$

$$t_f^v=-\sum _{g\in G}(v{)}_g^f{t}_g,f\in F_v,$$

$${{\mu }_v}^{-1}(f)=\{z\in U_v\mid z_f=0\}$$

$$T/\exp (\mathbb{R}{X}_f),$$

$$[{\sigma }_M]=\sum _{f\in F}{\lambda }_f{c}_f,$$

$$(t\cdot z{)}_j=e^{2\pi \text{mbox}\mathrm{i}\iota (t{)}_j}{z}_j,1\le j\le n.$$

$$U_{v,w}$$

$${\phi }_w\circ {{\phi }_v}^{-1}={{\psi }_w}^{-1}\circ {\psi }_v:U_{v,w}\to U_{w,v},$$

$$X_f=\sum _{g\in F_w}(w{)}_f^g{X}_g.$$

$$z_g^w=\prod _{f\in F_v}(z_f^v{)}^{(w{)}_f^g}/\prod _{f\in F_v\setminus F_w}|z_f^v{|}^{(w{)}_f^g}$$

$$z_g^w=\prod _{f\in F_v}(z_f^v{)}^{(w{)}_f^g}{r}_g({\mu }_v(z^v))/\prod _{f\in F_v\setminus F_w}|z_f^v{|}^{(w{)}_f^g}$$

$$s_w(z^w{)}_f=e^{\text{mbox}\mathrm{i}{t}_f}{s}_v(z^v{)}_f\text{mbox}\mathrm{forevery}f\in F$$

Full text

1 Introduction

Let $(M,\,\sigma)$ be a smooth compact and connected symplectic manifold of dimension $2n$ and let $T$ be a torus which acts effectively on $(M,\,\sigma)$ by means of symplectomorphisms. If the action of $T$ on $(M,\,\sigma)$ is moreover Hamiltonian, then $\!\!\mathop{\rm~{}dim}\nolimits T\leq n$, and the image of the momentum mapping $\mu_{T}:M\to{\mathfrak{t}}^{*}$ is a convex polytope $\Delta$ in the dual space $\mathfrak{t}^{*}$ of $\mathfrak{t}$, where $\mathfrak{t}$ denotes the Lie algebra of $T$. In the maximal case when $\!\!\mathop{\rm~{}dim}\nolimits T=n$, $(M,\,\sigma)$ is called a Delzant space.

Delzant [3, (*) on p. 323] proved that in this case the polytope $\Delta$ 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 ${\mathfrak{t}}^{*}$. Thirdly Delzant [3, pp. 328, 329] proved that for every Delzant polytope $\Delta$ there exists a Delzant space such that $\mu_{T}(M)=\Delta$. 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 $\lambda_{N}$ of the momentum mapping of the Hamiltonian $N$-action, where $E$, $N$ and $\lambda_{N}$ 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 $\Delta$ is $T$-equivariantly diffeomorphic to the toric variety $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ defined by the fan. Here $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ is a complex $n$-dimensional complex analytic manifold, and the action of the real torus $T$ on $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ has an extension to a complex analytic action on $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ of the complexification $T_{{\mathbb{C}}}$ of $T$. In our description in Section 5 of the toric variety $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ we do not use fans. The information, for each vertex $v$ of $\Delta$, which codimension one faces of $\Delta$ contain $v$, already suffices to define $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$.

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 $\varphi_{v}$ of a $T$-invariant open cell $M_{v}$ in $M$, where $M_{v}$ contains the unique fixed point $m_{v}$ in $M$ of the $T$-action such that $\mu_{T}(m_{v})=v$. We give an explicit construction of the inverse $\psi_{v}$ of $\varphi_{v}$, which is a maximal diffeomorphism in the sense of Remark 3.7. The construction of $\psi_{v}$ originated in an attempt to extend the equivariant symplectic ball embeddings from $({\mathbb{B}}^{2n}_{r},\,\sigma_{0})\subset({\mathbb{C}}^{n},\,\sigma_{0})$ into the Delzant space $(M,\,\sigma)$ in Pelayo [10] by maximal equivariant symplectomorphisms from open neighborhoods of the origin in ${\mathbb{C}}^{n}$ into the Delzant space $(M,\,\sigma)$. If $v$ and $w$ are two different vertices, then the coordinate transformation $\varphi_{w}\circ{\varphi_{v}}^{-1}$ is given by the explicit formulas (4.3), (4.4).

Let $\Sigma$ be the set of all strata of the orbit type stratification of $M$ for the $T$-action. Then the domain of definition $M_{v}$ of $\varphi_{v}$ is equal to the union of all $S\in\Sigma$ such that the fixed point $m_{v}$ belongs to the closure of $S$ in $M$, see Corollary 5.5. The strata $S\in\Sigma$ are also the orbits in the toric variety $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}\simeq M$ for the action of the complexification $T_{{\mathbb{C}}}$ of the real torus $T$, and the domain of definition $M_{v}$ of $\varphi_{v}$ is equal to the domain of definition of a natural complex analytic $T_{{\mathbb{C}}}$-equivariant coordinatization $\Phi_{v}$ of a $T_{{\mathbb{C}}}$-invariant open cell. The diffeomorphism $\Phi_{v}\circ{\varphi_{v}}^{-1}$, which sends the reduced phase space coordinates to the toric variety coordinates, maps $U_{v}:=\varphi_{v}(M_{v})$ 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 transformations 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 $\Delta$ and, for every vertex $v$ of $\Delta$, let $F_{v}$ denote the set of all $f\in F$ such that $v\in f$. Note that $\#(F_{v})=n$ for every vertex $v$ of $\Delta$. For any sets $A$ and $B$, let $A^{B}$ denote the set of all $A$-valued functions on $B$. If $A$ is a field and the set $B$ is finite, then $A^{B}$ 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 ${\mathbb{C}}^{F}$, and letting, for each vertex $v$, the coordinatizations $\varphi_{v}$ and $\Phi_{v}$ take their values in ${\mathbb{C}}^{F_{v}}$. This leads to a natural projection $\rho_{v}:{\mathbb{C}}^{F}\to{\mathbb{C}}^{F_{v}}$ obtained by the restriction of functions on $F$ to $F_{v}\subset F$. For each vertex $v$ the complex vector space ${\mathbb{C}}^{F_{v}}$ is isomorphic to ${\mathbb{C}}^{n}$, but the isomorphism depends on an enumeration of $F_{v}$, the introduction of which would lead to an unnecessary complication of the combinatorics. Similarly our torus $T$ is isomorphic to ${\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$, but the isomorphism depends on the choice of a ${\mathbb{Z}}$-basis of the integral lattice ${\mathfrak{t}}_{{\mathbb{Z}}}$ in the Lie algebra ${\mathfrak{t}}$ of $T$. As for each vertex $v$ a different ${\mathbb{Z}}$-basis of ${\mathfrak{t}}_{{\mathbb{Z}}}$ 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 ${\mathbb{C}}^{F}$, each ${\mathbb{C}}^{F_{v}}$, and $T$ is denoted as ${\mathbb{C}}^{d}$, ${\mathbb{C}}^{n}$, and ${\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$, 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 ${\mathfrak{t}}$. It follows that the exponential mapping $\!\!\mathop{\rm~{}exp}\nolimits:{\mathfrak{t}}\to T$ is a surjective homomorphism from the additive Lie group ${\mathfrak{t}}$ onto $T$. Furthermore, ${\mathfrak{t}}_{{\mathbb{Z}}}:=\!\!\mathop{\rm~{}ker}\nolimits(\!\!\mathop{\rm~{}exp}\nolimits)$ is a discrete subgroup of $({\mathfrak{t}},\,+)$ such that the exponential mapping induces an isomorphism from ${\mathfrak{t}}/{\mathfrak{t}}_{{\mathbb{Z}}}$ onto $T$, which we also denote by $\!\!\mathop{\rm~{}exp}\nolimits$. Note that ${\mathfrak{t}}_{{\mathbb{Z}}}$ is defined in terms of the group $T$ rather than only the Lie algebra ${\mathfrak{t}}$, but the notation ${\mathfrak{t}}_{{\mathbb{Z}}}$ has the advantage over the more precise notation $T_{{\mathbb{Z}}}$ that it reminds us of the fact it is a subgroup of the additive group ${\mathfrak{t}}$.

Because ${\mathfrak{t}}/{\mathfrak{t}}_{{\mathbb{Z}}}$ is compact, ${\mathfrak{t}}_{{\mathbb{Z}}}$ has a ${\mathbb{Z}}$-basis which at the same time is an ${\mathbb{R}}$-basis of ${\mathfrak{t}}$, and each ${\mathbb{Z}}$-basis of ${\mathfrak{t}}_{{\mathbb{Z}}}$ is an ${\mathbb{R}}$-basis of ${\mathfrak{t}}$. Using coordinates with respect to an ordered ${\mathbb{Z}}$-basis of ${\mathfrak{t}}_{{\mathbb{Z}}}$, we obtain a linear isomorphism from ${\mathfrak{t}}$ onto ${\mathbb{R}}^{n}$ which maps ${\mathfrak{t}}_{{\mathbb{Z}}}$ onto ${\mathbb{Z}}^{n}$, and therefore induces an isomorphism from $T$ onto ${\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$. For this reason, ${\mathfrak{t}}_{{\mathbb{Z}}}$ is called the integral lattice in ${\mathfrak{t}}$. However, because we do not have a preferred ${\mathbb{Z}}$-basis of ${\mathfrak{t}}_{{\mathbb{Z}}}$, we do not write $T={\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$.

Let $\Delta$ be an $n$-dimensional convex polytope in ${\mathfrak{t}}^{*}$. We denote by $F$ and $V$ the set of all codimension one faces and vertices of $\Delta$, 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 $\Delta$ is compact. For every $v\in V$, we write

[TABLE]

$\Delta$ is called a Delzant polytope if it has the following properties, see Guillemin [7, p. 8].

• i)

For each $f\in F$ there is an $X_{f}\in{\mathfrak{t}}$ and $\lambda_{f}\in{\mathbb{R}}$ such that the hyperplane which contains $f$ is equal to the set of all $\xi\in{\mathfrak{t}}^{*}$ such that $\langle X_{f},\,\xi\rangle+\lambda_{f}=0$, and $\Delta$ is contained in the set of all $\xi\in{\mathfrak{t}}^{*}$ such that $\langle X_{f},\,\xi\rangle+\lambda_{f}\geq 0$. The vector $X_{f}$ and constant $\lambda_{f}$ are made unique by requiring that they are not an integral multiple of another such vector and constant, respectively.

• ii)

For every $v\in V$, the $X_{f}$ with $f\in F_{v}$ form a ${\mathbb{Z}}$-basis of the integral lattice ${\mathfrak{t}}_{{\mathbb{Z}}}$ in ${\mathfrak{t}}$.

It follows that

[TABLE]

Also, $\#(F_{v})=n$ for every $v\in V$, which already makes the polytope $\Delta$ quite special. In the sequel we assume that $\Delta$ is a given Delzant polytope in ${\mathfrak{t}}^{*}$.

For any $z\in{\mathbb{C}}^{F}$ and $f\in F$ we write $z(f)=z_{f}$, which we view as the coordinate of the vector $z$ with the index $f$. Let $\pi$ be the real linear map from ${\mathbb{R}}^{F}$ to ${\mathfrak{t}}$ defined by

[TABLE]

Because, for any vertex $v$, the $X_{f}$ with $f\in F_{v}$ form a ${\mathbb{Z}}$-basis of ${\mathfrak{t}}_{{\mathbb{Z}}}$ which is also an ${\mathbb{R}}$-basis of ${\mathfrak{t}}$, we have $\pi({\mathbb{Z}}^{F})={\mathfrak{t}}_{{\mathbb{Z}}}$ and $\pi({\mathbb{R}}^{F})={\mathfrak{t}}$. It follows that $\pi$ induces a surjective homomorphism of Lie groups $\pi^{\prime}$ from the torus ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}=({\mathbb{R}}/{\mathbb{Z}})^{F}$ onto ${\mathfrak{t}}/{\mathfrak{t}}_{{\mathbb{Z}}}$, and we have the corresponding surjective homomorphism $\!\!\mathop{\rm~{}exp}\nolimits\circ\pi^{\prime}$ from ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ onto $T$.

Write ${\mathfrak{n}}:=\!\!\mathop{\rm~{}ker}\nolimits\pi$, a linear subspace of ${\mathbb{R}}^{F}$, and $N=\!\!\mathop{\rm~{}ker}\nolimits(\!\!\mathop{\rm~{}exp}\nolimits\circ\pi^{\prime})$, a compact commutative subgroup of the torus ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$. Actually, $N$ is connected, see Lemma 3.1 below, and therefore isomorphic to ${\mathfrak{n}}/{\mathfrak{n}}_{{\mathbb{Z}}}$, where ${\mathfrak{n}}_{{\mathbb{Z}}}:={\mathfrak{n}}\cap{\mathbb{Z}}^{F}$ is the integral lattice in ${\mathfrak{n}}$ of the torus $N$. 111We did not find a proof of the connectedness of $N$ in [3], [2], or [7].

On the complex vector space ${\mathbb{C}}^{F}$ of all complex-valued functions on $F$ we have the action of the torus ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$, where $t\in{\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ maps $z\in{\mathbb{C}}^{F}$ to the element $t\cdot z\in{\mathbb{C}}^{F}$ defined by

[TABLE]

The infinitesimal action of $Y\in{\mathbb{R}}^{F}=\!\!\mathop{\rm~{}Lie}\nolimits({\mathbb{R}}^{F}/{\mathbb{Z}}^{F})$ is given by

[TABLE]

which is a Hamiltonian vector field defined by the function

[TABLE]

and with respect to the symplectic form

[TABLE]

if $z_{f}=x_{f}+\!\!\mathop{\rm~{}i}\nolimits y_{f}$, with $x_{f},\,y_{f}\in{\mathbb{R}}$. Here the factor $1/2\pi$ is introduced in order to avoid an integral lattice $(2\pi\,{\mathbb{Z}})^{F}$ instead of our ${\mathbb{Z}}^{F}$.

Because the right hand side of (2.3) depends linearly on $Y$, we can view $\mu(z)$ as an element of $({\mathbb{R}}^{F})^{*}\simeq{\mathbb{R}}^{F}$, with the coordinates

[TABLE]

In other words, the action of ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ on ${\mathbb{C}}^{F}$ is Hamiltonian, with respect to the symplectic form $\sigma$ and with momentum mapping $\mu:{\mathbb{C}}^{F}\to(\!\!\mathop{\rm~{}Lie}\nolimits({\mathbb{R}}^{F}/{\mathbb{Z}}^{F}))^{*}$ given by (2.3), or equivalently (2.5).

It follows that the subtorus $N$ of ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ acts on ${\mathbb{C}}^{F}$ in a Hamiltonian fashion, with momentum mapping

[TABLE]

where $\iota_{\mathfrak{n}}:{\mathfrak{n}}\to{\mathbb{R}}^{F}$ denotes the identity viewed as a linear mapping from ${\mathfrak{n}}\subset{\mathbb{R}}^{F}$ to ${\mathbb{R}}^{F}$, and its transposed $\iota_{\mathfrak{n}}^{*}:({\mathbb{R}}^{F})^{*}\to{\mathfrak{n}}^{*}$ is the map which assigns to each linear form on ${\mathbb{R}}^{F}$ its restriction to ${\mathfrak{n}}$.

Write $\lambda_{N}=\iota_{\mathfrak{n}}^{*}(\lambda)$, where $\lambda$ denotes the element of $({\mathbb{R}}^{F})^{*}\simeq{\mathbb{R}}^{F}$ with the coordinates $\lambda_{f}$, $f\in F$. It follows from Guillemin [7, Th. 1.6 and Th. 1.4] that $\lambda_{N}$ is a regular value of $\mu_{N}$, hence the level set $Z:={\mu_{N}}^{-1}(\{\lambda_{N}\})$ of $\mu_{N}$ for the level $\lambda_{N}$ is a smooth submanifold of ${\mathbb{C}}^{F}$, and that the action of $N$ on $Z$ is proper and free. As a consequence the $N$-orbit space $M=M_{\Delta}:=Z/N$ is a smooth $2n$-dimensional manifold such that the projection $p:Z\to M$ exhibits $Z$ as a principal $N$-bundle over $M$. Moreover, there is a unique symplectic form $\sigma_{M}$ on $M$ such that $p^{*}\sigma_{M}={\iota_{Z}}^{*}\sigma$, where $\iota_{Z}$ is the identity viewed as a smooth mapping from $Z$ to ${\mathbb{C}}^{F}$.

**Remark 2.1  ** Guillemin [7] used the momentum mapping $\mu_{N}-\lambda_{N}$ instead of $\mu_{N}$, 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 $\mu_{N}$ for the $N$-action, which does not depend on $\lambda$, and do the reduction at the level $\lambda_{N}$.

$\oslash$

The symplectic manifold $(M,\,\sigma_{M})$ is the Marsden-Weinstein reduction of the symplectic manifold $({\mathbb{C}}^{F},\,\sigma)$ for the Hamiltonian $N$-action at the level $\lambda_{N}$ 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 $({\mathbb{R}}^{F}/{\mathbb{Z}}^{F})/N\simeq T$, with momentum mapping $\mu_{T}:M\to{\mathfrak{t}}^{*}$ determined by

[TABLE]

The torus $T$ acts effectively on $M$ and $\mu_{T}(M)=\Delta$, 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 $z_{f}$, $f\in F$.

The symplectic manifold $M_{\Delta}$ together with this Hamiltonian $T$-action is called the Delzant space defined by $\Delta$, 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 $v\in V$, let $\iota_{v}:={\rho_{v}}^{*}:({\mathbb{R}}^{F_{v}})^{*}\to({\mathbb{R}}^{F})^{*}$ denote the transposed of the restriction projection $\rho_{v}:{\mathbb{R}}^{F}\to{\mathbb{R}}^{F_{v}}$. If in the usual way we identify $({\mathbb{R}}^{F_{v}})^{*}$ and $({\mathbb{R}}^{F})^{*}$ with ${\mathbb{R}}^{F_{v}}$ and ${\mathbb{R}}^{F}$, respectively, then $\iota_{v}:{\mathbb{R}}^{F_{v}}\to{\mathbb{R}}^{F}$ is the embedding defined by $\iota_{v}(x)_{f}=x_{f}$ if $f\in F_{v}$ and $\iota_{v}(x)_{f^{\prime}}=0$ if $f^{\prime}\in F$, $f^{\prime}\notin F_{v}$. Because $\iota_{v}$ maps ${\mathbb{Z}}^{F_{v}}$ into ${\mathbb{Z}}^{F}$ and $\iota_{v}({\mathbb{R}}^{F_{v}})\cap{\mathbb{Z}}^{F}=\iota_{v}({\mathbb{Z}}^{F_{v}})$, it induces an embedding of the $n$-dimensional torus ${\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$ into ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$, which we also denote by $\iota_{v}$.

Lemma 3.1

With these notations, ${\mathbb{R}}^{F}$, ${\mathbb{Z}}^{F}$, and ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ are the direct sum of ${\mathfrak{n}}$ and $\iota_{v}({\mathbb{R}}^{F_{v}})$, ${\mathfrak{n}}\cap{\mathbb{Z}}^{n}$ and $\iota_{v}({\mathbb{Z}}^{F_{v}})$, and $N$ and $\iota_{v}({\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}})$, respectively.

It follows that $N$ is connected, a torus, with integral lattice equal to ${\mathfrak{n}}\cap{\mathbb{Z}}^{F}$. It also follows that $\pi\circ\iota_{v}$ is an isomorphism from the torus ${\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$ onto the torus $T$.

Proof   Let $t\in{\mathbb{R}}^{F}$. Because the $X_{f}$, $f\in F_{v}$, form an ${\mathbb{R}}$-basis of ${\mathfrak{t}}$, there exists a unique $t^{v}\in{\mathbb{R}}^{F_{v}}$, such that

[TABLE]

that is, $t-\iota_{v}(t^{v})\in{\mathfrak{n}}$. Moreover, because the $X_{f}$, $f\in F_{v}$, also form a ${\mathbb{Z}}$-basis of ${\mathfrak{t}}_{{\mathbb{Z}}}$, we have that $t^{v}\in{\mathbb{Z}}^{F_{v}}$, and therefore $t-\iota_{v}(t^{v})\in{\mathfrak{n}}\cap{\mathbb{Z}}^{F}$, if $t\in{\mathbb{Z}}^{F}$.

$\Box$

Lemma 3.2

We have $z\in Z$ if and only if $\mu(z)-\lambda\in\pi^{*}({\mathfrak{t}}^{*})$. More explicitly, if and only if there exists a $\xi\in{\mathfrak{t}}^{*}$ such that

[TABLE]

When $z\in Z$, the $\xi$ in (3.1) is uniquely determined.

Furthermore, $Z=(\mu-\lambda)^{-1}(\pi^{*}(\Delta))$, $(\mu-\lambda)(Z)=\pi^{*}(\Delta)$, and $Z$ is a compact subset of ${\mathbb{C}}^{F}$.

Proof   The kernel of $\iota_{\mathfrak{n}}^{*}$ is equal to the space of all linear forms on ${\mathbb{R}}^{F}$ which vanish on ${\mathfrak{n}}:=\!\!\mathop{\rm~{}ker}\nolimits\pi$, and therefore $\!\!\mathop{\rm~{}ker}\nolimits\iota_{\mathfrak{n}}^{*}$ is equal to the image of $\pi^{*}:{\mathfrak{t}}^{*}\to({\mathbb{R}}^{F})^{*}$. Because $\pi$ is surjective, $\pi^{*}$ is injective, which proves the uniqueness of $\xi$.

It follows from (3.1) that $\langle X_{f},\,\xi\rangle+\lambda_{f}\geq 0$ for every $f\in F$, and therefore $\xi\in\Delta$ in view of (2.1). Conversely, if $\xi\in\Delta$, then there exists for every $f\in F$ a complex number $z_{f}$ such that $|z_{f}|^{2}/2=\langle X_{f},\,\xi\rangle+\lambda_{f}$, which means that $z\in Z$ and $(\mu-\lambda)(z)=\pi^{*}(\xi)$. The set $\pi^{*}(\Delta)$ is compact because $\Delta$ is compact and $\pi^{*}$ is continuous. Because the mapping $\mu-\lambda$ is proper, it follows that $Z=(\mu-\lambda)^{-1}(\pi^{*}(\Delta))$ is compact.

$\Box$

Let $v\in V$. The $X_{f}$, $f\in F_{v}$, form an ${\mathbb{R}}$-basis of ${\mathfrak{t}}$, and therefore there exists for each $z\in{\mathbb{C}}^{F_{v}}$ a unique $\xi=\mu_{v}(z)\in{\mathfrak{t}}^{*}$ such that (3.1) holds for every $f\in F_{v}$. That is, the mapping $\mu_{v}:{\mathbb{C}}^{F_{v}}\to{\mathfrak{t}}^{*}$ is defined by the equations

[TABLE]

In other words, $\mu_{v}$ is defined by the formula

[TABLE]

where $\rho_{v}$ denotes the restriction projection from ${\mathbb{R}}^{F}$ onto ${\mathbb{R}}^{F_{v}}$.

Lemma 3.3

If we let $T$ act on ${\mathbb{C}}^{F_{v}}$ via ${\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$ by means of $(t,\,z)\mapsto(\pi\circ\iota_{v})^{-1}(t)\cdot z$, then $\mu_{v}:{\mathbb{C}}^{F_{v}}\to{\mathfrak{t}}^{*}$ is a momentum mapping for this Hamiltonian action of $T$ on ${\mathbb{C}}^{F_{v}}$, with $\mu_{v}(0)=v$. Here the symplectic form on ${\mathbb{C}}^{F_{v}}$ is equal to

[TABLE]

that is, (2.4) with $F$ replaced by $F_{v}$.

Let $\rho_{v}$ denote the restriction projection from ${\mathbb{C}}^{F}$ onto ${\mathbb{C}}^{F_{v}}$, and let $U_{v}$ be the interior of the subset $\rho_{v}(Z)$ of ${\mathbb{C}}^{F_{v}}$. Write

[TABLE]

Then $\rho_{v}(Z)={\mu_{v}}^{-1}(\Delta)$, $\mu_{v}(\rho_{v}(Z))=\Delta$, $U_{v}={\mu_{v}}^{-1}(\Delta_{v})$, and $\mu_{v}(U_{v})=\Delta_{v}$. In particular $\rho_{v}(Z)$ is a compact subset of ${\mathbb{C}}^{F_{v}}$, and $U_{v}$ is a bounded and connected open neighborhood of [math] in ${\mathbb{C}}^{F_{v}}$.

Proof   The first statement follows from (3.3), the fact that $\rho_{v}\circ\mu\circ\iota_{v}$ is a momentum mapping for the standard ${\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$ action on ${\mathbb{C}}^{F_{v}}$, 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 $\langle X_{f},\,\mu_{v}(0)\rangle+\lambda_{f}=0$ for every $f\in F_{v}$, hence $\mu_{v}(0)=v$ in view of i) in the definition of a Delzant polytope, and the fact that $\{v\}$ is the intersection of all the $f\in F_{v}$.

It follows from (3.2), Lemma 3.2, that $z\in Z$ if and only if

[TABLE]

where we note that these equations are satisfied by definition for the $f\in F_{v}$. Therefore, if $z\in Z$, then (3.6) and (2.1) imply that $\mu_{v}(\rho(z))\in\Delta$. Conversely, if $\xi\in\Delta$, then it follows from Lemma 3.2 that there exists $z\in Z$ such that $\pi^{*}(\xi)=\mu(z)-\lambda$, of which the restriction to $F_{v}$ yields $\xi=\mu_{v}(\rho(z))$.

If $\xi\in\Delta_{v}$, $z^{v}\in{\mathbb{C}}^{F_{v}}$, $\mu_{v}(z^{v})=\xi$, then $\langle X_{f^{\prime}},\,\mu_{v}(z^{v})\rangle+\lambda_{f^{\prime}}>0$ for every $f^{\prime}\in F\setminus F_{v}$, which will remain valid if we replace $z^{v}$ by $\widetilde{z}^{v}$ in a sufficiently small neighborhood of $z^{v}$ in ${\mathbb{C}}^{F_{v}}$. It follows that we can find $\widetilde{z}\in{\mathbb{C}}^{F}$ such that $\rho_{v}(\widetilde{z})=\widetilde{z}^{v}$ and (3.6) holds with $z$ replaced by $\widetilde{z}$. That is, $\widetilde{z}\in Z$, and we have proved that $z^{v}\in U_{v}$.

Let conversely $z\in U_{v}\subset{\mathbb{C}}^{F_{v}}$. We have in view of (3.2) that

[TABLE]

for every $f\in F_{v}$. Therefore $\mu_{v}(z)-v$ is multiplied by $c^{2}$ if we replace $z$ by $c\,z$, $c>0$. Because $z$ is in the interior of $\rho_{v}(Z)$, we have $c\,z^{v}\in\rho_{v}(Z)$, hence $\mu_{v}(c\,z)\in\Delta$ for $c>1$, $c$ sufficiently close to $1$. On the other hand, if $\xi$ belongs to a face of $\Delta$ which is not adjacent to $v$, then $v+\tau\,(\xi-v)\notin\Delta$ for any $\tau>1$. It follows that $\mu_{v}(z)$ does not belong to any $f^{\prime}\in F\setminus F_{v}$, that is, $\mu_{v}(z)\in\Delta_{v}$.

$\Box$

The equation (3.6) can be written in the form $|z_{f}|=r_{f}(\mu_{v}(\rho_{v}(z)))$, where, for each $f\in F$, the function $r_{f}:\Delta\to{\mathbb{R}}_{\geq 0}$ is defined by

[TABLE]

We now view the equations (3.6) for $z\in Z$ as equations for the coordinates $z_{f^{\prime}}$, $f^{\prime}\in F\setminus F_{v}$, with the $z_{f}$, $f\in F_{v}$ as parameters, where the latter constitute the vector $z^{v}=\rho_{v}(z)$. If $z^{v}\in U_{v}$, then for each $f^{\prime}\in F\setminus F_{v}$ the coordinate $z_{f^{\prime}}$ lies on the circle about the origin with strictly positive radius $r_{f^{\prime}}(\mu_{v}(z^{v})$. Because Lemma 3.1 implies that the homomorphism which assigns to each element of $N$ its projection to ${\mathbb{R}}^{F\setminus F_{v}}/{\mathbb{Z}}^{F\setminus F_{v}}$ is an isomorphism, and the latter torus is the group of the coordinatewise rotations of the $z_{f^{\prime}}$, $f^{\prime}\in F\setminus F_{v}$, this leads to the following conclusions.

Proposition 3.4

Let $v$ be a vertex of $\Delta$. The open subset $Z_{v}:={\rho_{v}}^{-1}(U_{v})\cap Z$ of $Z$ is a connected smooth submanfold of ${\mathbb{C}}^{F}$ of real dimension $2n+(d-n)$, where $d=\#(F)$ and $d-n=\!\!\mathop{\rm~{}dim}\nolimits N$. The action of the torus $N$ on $Z_{v}$ is free, and the projection $\rho_{v}:Z_{v}\to U_{v}$ exhibits $Z_{v}$ as a principal $N$-bundle over $U_{v}$. It follows that we have a reduced phase space $M_{v}:=Z_{v}/N$, 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 $Z_{v}$.

There is a unique global section $s_{v}:U_{v}\to Z_{v}$ of $\rho_{v}:Z_{v}\to U_{v}$ such that $s_{v}(z)_{f^{\prime}}\in{\mathbb{R}}_{>0}$ for every $z\in U_{v}$ and $f^{\prime}\in F\setminus F_{v}$. Actually, $s_{v}(z)_{f^{\prime}}=r_{f^{\prime}}(\mu_{v}(z))$ when $z\in U_{v}$ and $f^{\prime}\in F\setminus F_{v}$, and therefore the section $s_{v}$ is smooth. If $p_{v}:Z_{v}\to M_{v}=Z_{v}/N$ denotes the canonical projection, then $\psi_{v}:=p_{v}\circ s_{v}$ is a $T$-equivariant symplectomorphism from $U_{v}$ onto $M_{v}$, where $T$ acts on $U_{v}$ via ${\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$, as in Lemma 3.3.

**Remark 3.5  ** When $z$ belongs to the closure $\rho_{v}(Z)=\overline{U_{v}}$ of $U_{v}$ in ${\mathbb{C}}^{F_{v}}$, see Lemma 3.3, we can define $s_{v}(z)\in{\mathbb{C}}^{F}$ by $s_{v}(z)_{f}=z_{f}$ when $f\in F_{v}$ and $s_{v}(z)_{f^{\prime}}=r_{f^{\prime}}(\mu_{v}(z))$ when $f^{\prime}\in F\setminus F_{v}$. This defines a continuous extension $s_{v}:\overline{U_{v}}\to{\mathbb{C}}^{F}$ of the mapping $s_{v}:U_{v}\to Z$. Therefore $s_{v}(\overline{U_{v}})\subset Z$, and $\psi_{v}:=p\circ s_{v}:\overline{U_{v}}\to M$ is a continuous extension of the diffeomorphism $\psi_{v}:U_{v}\to M_{v}$.

The continuous mapping $\psi_{v}:\overline{U_{v}}\to M$ is surjective, but the restriction of it to the boundary $\partial U_{v}:=\overline{U_{v}}\setminus U_{v}$ of $U_{v}$ in ${\mathbb{C}}^{F_{v}}$ is not injective. If $z^{v}\in\partial U_{v}$, then the set $G$ of all $f^{\prime}\in F\setminus F_{v}$ such that $s_{v}(z^{v})_{f^{\prime}}=0$, or equivalently $\mu_{T}(\psi_{v}(z^{v}))\in f^{\prime}$, is not empty. The fiber of $\psi_{v}$ over $\psi_{v}(z^{v})$ is equal to the set of all $t^{v}\cdot z^{v}$, where the $t^{v}\in{\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$ are of the form

[TABLE]

where $t_{g}\in{\mathbb{R}}/{\mathbb{Z}}$. It follows that each fiber is an orbit of some subtorus of ${\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$ acting on ${\mathbb{C}}^{F_{v}}$.

$\oslash$

Recall the definition (3.5) of the open subset $\Delta_{v}$ of the Delzant polytope $\Delta$. Because the union over all vertices $v$ of the $\Delta_{v}$ is equal to $\Delta$, we have the following corollary.

Corollary 3.6

The sets $Z_{v}$, $v\in V$, form a covering of $Z$. As a consequence, $Z$ is a smooth submanifold of ${\mathbb{C}}^{F}$ 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 $\mu_{T}:M\to T$ as in (2.7). The sets $M_{v}$, $v\in V$, form an open covering of $M$ and the $\varphi_{v}:=(\psi_{v})^{-1}:M_{v}\to U_{v}$ form an atlas of $T$-equivariant symplectic coordinatizations of the Hamiltonian $T$-space $M$. For each $v\in V$, we have $M_{v}={\mu_{T}}^{-1}(\Delta_{v})$, and $\mu_{T}|_{M_{v}}=\mu_{v}\circ\varphi_{v}$.

For a characterization of $M_{v}$ in terms of the orbit type stratification in $M$ for the $T$-action, see Corollary 5.5, which also implies that $M_{v}$ is an open cell in $M$.

Corollary 3.7

For every $f\in F$ the set ${\mu_{T}}^{-1}(f)$ is a real codimension two smooth compact connected smooth symplectic submanifold of $M$.

For each $v\in V$, the set $M_{v}$ is dense in $M$, and the diffeomorphism $\psi_{v}:U_{v}\to M_{v}$ is maximal among all diffeomorphisms from open subsets of ${\mathbb{C}}^{F_{v}}$ onto open subsets of $M$.

Proof   If $f\in F$, then for each $v\in V$ we have that

[TABLE]

if $v\in f$, that is, $f\in\Delta_{v}$. This follows from (3.2) and i) in the description of $\Delta$ in the beginning of Section 2. On the other hand, ${\mu_{v}}^{-1}(f)=\emptyset$ if $f\notin\Delta_{v}$. Because ${\mu_{T}}^{-1}(f)\cap M_{v}=\psi_{v}({\mu_{v}}^{-1}(f))$, and the $M_{v}$, $v\in V$, form an open covering of $M$, this proves the first statement. The second statement follows from the first one, because the complement of $M_{v}$ in $M$ is equal to the union of the sets ${\mu_{T}}^{-1}(f^{\prime})$ with $f^{\prime}\in F\setminus F_{v}$.

$\Box$

**Remark 3.8  ** It follows from the proof of Corollary 3.7, that ${\mu_{T}}^{-1}(f)$ is a connected component of the fixed point set in $M$ of the of the circle subgroup $\!\!\mathop{\rm~{}exp}\nolimits({\mathbb{R}}\,X_{f})$ of $T$.

Actually, ${\mu_{T}}^{-1}(f)$ is a Delzant space for the action of the $(n-1)$-dimensional torus

[TABLE]

with Delzant polytope $P\subset({\mathfrak{t}}/({\mathbb{R}}\,X_{f}))^{*}$ such that the image of $P$ in ${\mathfrak{t}}^{*}$ under the embedding $({\mathfrak{t}}/({\mathbb{R}}\,X_{f}))^{*}\to{\mathfrak{t}}^{*}$ is equal to a translate of $f$.

In a similar way, if $g$ is a $k$-dimensional face of $\Delta$, then ${\mu_{T}}^{-1}(g)$ is a $2k$-dimensional Delzant space for the quotient of $T$ by the subtorus of $T$ which acts trivially on ${\mu_{T}}^{-1}(g)$.

$\oslash$

**Remark 3.9  ** Let, for each $f\in F$, $c_{f}\in\!\!\mathop{\rm~{}H}\nolimits^{2}(M,\,{\mathbb{Z}})\subset\!\!\mathop{\rm~{}H}\nolimits^{2}(M,\,{\mathbb{R}})$ denote the Poincaré dual of the codimension two Delzant subspace ${\mu_{T}}^{-1}(f)$ of $M$, see Remark 3.7. Then, with our normalization of the symplectic form (2.4), the de Rham cohomology class $[\sigma_{M}]$ of the symplectic form $\sigma_{M}$ of the Delzant space is equal to

[TABLE]

see Guillemin [8, Thm. 6.3]. In particular $[\sigma_{M}]\in\!\!\mathop{\rm~{}H}\nolimits^{2}(M,\,{\mathbb{Z}})$, and therefore $[\sigma_{M}]$ is equal to the Chern class of a complex line bundle over $M$, if all the coefficients $\lambda_{f}$, $f\in F$, are integers.

If $\Delta$ is a simplex, when $M$ is isomorphic to the $n$-dimensional complex projective space, then the ${\mu_{T}}^{-1}(f)$, $f\in F$, are complex projective hyperplanes, see Subsection 6.1, which are all homologous to each other. It follows that in this case $[\sigma_{M}]=\gamma\,c$, where $c$ is the Poincaré dual of a complex projective hyperplane and $\gamma$ is equal to the sum of all the coefficients $\lambda_{f}$, $f\in F$.

$\oslash$

**Remark 3.10  ** Let $\iota:T\to{\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$ be an isomorphism of tori, which allows us to let $t\in T$ act on ${\mathbb{C}}^{n}$ via ${\mathbb{R}}^{n}/{\mathbb{Z}}^{n}$ by means of

[TABLE]

Let $U$ be a connected $T$-invariant open neighborhood of [math] in ${\mathbb{C}}^{n}$, provided with the symplectic form (2.4) with $F$ replaced by $\{1,\,\ldots,\,n\}$. Let $\psi:U\to M$ be a $T$-equivariant symplectomorphism from $U$ onto an open subset $\psi(U)$ of $M$. Because [math] 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 $\mu_{T}$ of the vertices of $\Delta$, there is a unique $v\in V$ such that $\mu_{T}(\psi(0))=v$. Let $I_{v}:{\mathbb{C}}^{F_{v}}\to{\mathbb{C}}^{n}$ denote the complex linear extension of the tangent map of the torus isomorphism $\iota\circ(\pi\circ\iota_{v})$. In terms of the notation of Lemma 3.3 and Proposition 3.4, we have that $U\subset I_{v}(U_{v})$ and $\psi_{v}=\psi\circ I_{v}$ on ${I_{v}}^{-1}(U)$, which leads to an identification of $\psi$ with the restriction of $\psi_{v}$ to the connected open subset ${I_{v}}^{-1}(U)$ of $U_{v}$, via the isomorphism ${I_{v}}^{-1}$.

The $\psi$’s, with $U$ equal to a ball in ${\mathbb{C}}^{n}$ centered at the origin, are the equivariant symplectic ball embeddings in Pelayo [10], and the second statement in Corollary 3.7 shows that the diffeomorphisms $\psi_{v}$ are the maximal extensions of these equivariant symplectic ball embeddings.

$\oslash$

4 The coordinate transformations

Let $v,\,w\in V$. Then

[TABLE]

In this section we will give an explicit formula for the coordinate transformations

[TABLE]

which then leads to a description of the Delzant space $M$ as obtained by gluing together the subsets $U_{v}$ with the coordinate transformations as the gluing maps.

Let $f\in F$. Because the $X_{g}$, $g\in F_{w}$, form a ${\mathbb{Z}}$-basis of ${\mathfrak{t}}_{{\mathbb{Z}}}$, and $X_{f}\in{\mathfrak{t}}_{{\mathbb{Z}}}$, there exist unique integers $(w)_{f}^{g}$, $g\in F_{w}$, such that

[TABLE]

Note that if $f\in F_{w}$, then $(w)_{f}^{g}=1$ when $g=f$ and $(w)_{f}^{g}=0$ otherwise. For the following lemma recall that $r_{g}$ is defined by expression (3.7).

Lemma 4.1

Let $v,\,w\in V$, $z^{v}\in U_{v,\,w}$. Then $z^{w}:=\varphi_{w}\circ{\varphi_{v}}^{-1}(z^{v})\in U_{w}\subset{\mathbb{C}}^{F_{w}}$ is given by

[TABLE]

if $g\in F_{w}\cap F_{v}$, and

[TABLE]

if $g\in F_{w}\setminus F_{v}$.

Proof   The element $z^{w}\in U_{w}$ is determined by the condition that $s_{w}(z^{w})$ belongs to the $N$-orbit of $s_{v}(z^{v})$. That is,

[TABLE]

for some $t\in{\mathbb{R}}^{F}$ such that

[TABLE]

It follows from (4.5), (4.2) and the linear independence of the $X_{g}$, $g\in F_{w}$, that $t\in{\mathfrak{n}}$ if and only if

[TABLE]

Note that $\mu_{v}(z^{v})=\mu_{T}(m)=\mu_{w}(z^{w})$, where $m=\psi_{v}(z^{v})=\psi_{w}(z^{w})$. It follows from the definition of the sections $s_{v}$ and $s_{w}$, see Proposition 3.4, that

• i)

$s_{v}(z^{v})_{f}=z^{v}_{f}$ and $s_{w}(z^{w})_{f}=z^{w}_{f}$ if $f\in F_{v}\cap F_{w}$,

• ii)

$s_{v}(z^{v})_{f}=z^{v}_{f}$ and $s_{w}(z^{w})_{f}=r_{f}(\mu_{w}(z^{w}))=r_{f}(\mu_{v}(z^{v}))$ if $f\in F_{v}\setminus F_{w}$,

• iii)

$s_{v}(z^{v})_{f}=r_{f}(\mu_{v}(z^{v}))$ and $s_{w}(z^{w})_{f}=z^{w}_{f}$ if $f\in F_{w}\setminus F_{v}$, and

• iv)

$s_{v}(z^{v})_{f}=r_{f}(\mu_{v}(z^{v}))=r_{f}(\mu_{w}(z^{w}))=s_{w}(z^{w})_{f}$ if $f\in F\setminus(F_{v}\cup F_{w})$.

It follows from ii) and iv) that $t_{f}=\,-\!\!\mathop{\rm~{}arg}\nolimits z^{v}_{f}$ and $t_{f}=0$ modulo $2\pi$ if $f\in F_{v}\setminus F_{w}$ and $f\in F\setminus(F_{v}\cup F_{w})$, respectively. Then (4.6) implies that, modulo $2\pi$,

[TABLE]

It now follows from i) and iii) that if $g\in F_{w}$, then $z^{w}_{g}=s_{w}(z^{w})_{g}=\!\!\mathop{\rm~{}e}\nolimits^{\!\!\mathop{\mbox{\rm\footnotesize~{}i}}\nolimits\,t_{g}}\,s_{v}(z^{v})_{g}$ is equal to

[TABLE]

if $g\in F_{v}$, and equal to

[TABLE]

if $g\notin F_{v}$, respectively. Here we have used that if $g\in F_{w}$, then $(w)_{f}^{g}=1$ if $f=g$ and $(w)_{f}^{g}=0$ if $f\in F_{w}$, $f\neq g$. Because $|z^{v}_{g}|=r_{g}(\mu_{v}(z^{v}))$ if $g\notin F_{v}$, see (3.6) and (3.7), this completes the proof of the lemma.

$\Box$

**Remark 4.2  ** Note that $z^{v}\in U_{v,\,w}$ means that $z^{v}\in U_{v}$ and $z^{v}_{f}\neq 0$ if $f\in F_{v}\setminus F_{w}$. Furthermore, $z^{v}\in U_{v}$ implies that if $g\notin F_{v}$, then $\mu_{v}(z^{v})\notin g$, and therefore $r_{g}$ is smooth on a neighborhood of $\mu_{v}(z^{v})$. Finally, note that if $g\in F_{w}$ and $f\in F_{v}\cap F_{w}$, then $(w)_{f}^{g}\in\{0,\,1\}$, and therefore each of the factors in the right hand sides of (4.3) and (4.4) is smooth on $U_{v,\,w}$.

$\oslash$

**Remark 4.3  ** In (4.3) and (4.4) only the integers $(w)_{f}^{g}$ appear with $f\in F_{v}$ and $g\in F_{w}$. Let $(w\,v)$ denote the matrix $(w)_{f}^{g}$, where $f\in F_{v}$ and $g\in F_{w}$. 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 $u,\,v,\,w\in V$. These properties follow from the fact that (4.2) shows that $(w\,v)$ is the matrix which maps the ${\mathbb{Z}}$-basis $X_{g}$, $g\in F_{w}$, onto the ${\mathbb{Z}}$-basis $X_{f}$, $f\in F_{v}$, of ${\mathfrak{t}}_{{\mathbb{Z}}}$. It is no surprise that these base changes enter in the formulas which relate the models in the vector spaces ${\mathbb{C}}^{F_{v}}$ for the different choices of $v\in V$.

$\oslash$

Corollary 4.4

Let, for each $v\in V$, the mapping $\mu_{v}:{\mathbb{C}}^{F_{v}}\to{\mathfrak{t}}^{*}$ be defined by (3.2), which is a momentum mapping for a Hamiltonian $T$-action via ${\mathbb{R}}^{F_{v}}/{\mathbb{Z}}^{F_{v}}$ on the symplectic vector space ${\mathbb{C}}^{F_{v}}$ as in Lemma 3.3. Define $U_{v}:={\mu_{v}}^{-1}(\Delta_{v})$. If also $w\in V$, define $U_{v,\,w}$ as the right hand side of (4.1), and, if $z^{v}\in U_{v,\,w}$, define $\varphi_{w,\,v}(z^{v}):=z^{w}$, where $z^{w}\in{\mathbb{C}}^{F_{w}}$ is given by (4.3) and (4.4).

Then $\varphi_{w,\,v}$ is a $T$-equivariant symplectomorphism from $U_{v,\,w}$ onto $U_{w,\,v}$ such that $\mu_{w}=\mu_{v}\circ\varphi_{v,\,w}$ on $U_{w,\,v}$. The $\varphi_{w,\,v}$ satisfy the cocycle condition $\varphi_{w,\,v}\circ\varphi_{v,\,u}=\varphi_{w,\,u}$ where the left hand side is defined. Glueing together the Hamiltonian $T$-spaces $U_{v}$, $v\in V$, with the momentum maps $\mu_{v}$, by means of the gluing maps $\varphi_{w,\,v}$, $v,\,w\in V$, we obtain a compact connected smooth symplectic manifold $\widetilde{M}$ with an effective Hamiltonian $T$-action with a common momentum map $\widetilde{\mu}:\widetilde{M}\to T$ such that $\widetilde{\mu}(\widetilde{M})=\Delta$. In other words, $\widetilde{M}$ is a Delzant space for the Delzant polytope $\Delta$.

The Delzant space $\widetilde{M}$ is obviously isomorphic to the Delzant space $M=\mu^{-1}(\{\lambda\})/N$ introduced in Section 2, and actually the isomorphism is used in the proof that $\widetilde{M}$ is a Delzant space for the Delzant polytope $\Delta$. The only purpose of Corollary 4.4 is to exhibit the Delzant space as obtained from gluing together the $U_{v}$, $v\in V$, by means of the gluing maps $\varphi_{v,\,w}$, $v,\,w\in V$.

5 The toric variety

Let ${\mathbb{T}}:=\{z\in{\mathbb{C}}\mid|z|=1\}$ denote the unit circle in the complex plane. The mapping $t\mapsto u$ where $u_{f}=\!\!\mathop{\rm~{}e}\nolimits^{2\pi\!\!\mathop{\mbox{\rm\scriptsize~{}i}}\nolimits t_{f}}$ for every $f\in F$ is an isomorphism from the torus ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ onto ${\mathbb{T}}^{F}$, where ${\mathbb{T}}^{F}$ acts on ${\mathbb{C}}^{F}$ by means of coordinatewise multiplication and ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ acted on ${\mathbb{C}}^{F}$ via the isomorphism from ${\mathbb{R}}^{F}/{\mathbb{Z}}^{F}$ onto ${\mathbb{T}}^{F}$. The complexification ${\mathbb{T}}_{{\mathbb{C}}}$ of the compact Lie group ${\mathbb{T}}$ is the multiplicative group ${\mathbb{C}}^{\times}$ of all nonzero complex numbers, and the complexification of ${\mathbb{T}}^{F}$ is equal to ${\mathbb{T}}_{{\mathbb{C}}}^{F}:=({\mathbb{T}}_{{\mathbb{C}}})^{F}=({\mathbb{C}}^{\times})^{F}$, which also acts on ${\mathbb{C}}^{F}$ by means of coordinatewise multiplication.

The complexification $N_{{\mathbb{C}}}$ of $N$ is the subgroup $\!\!\mathop{\rm~{}exp}\nolimits({\mathfrak{n}}_{{\mathbb{C}}})$ of $U_{{\mathbb{C}}}^{F}$, where $\mathfrak{n}_{{\mathbb{C}}}:=\mathfrak{n}\oplus\!\!\mathop{\rm~{}i}\nolimits{\mathfrak{n}}\subset{\mathbb{C}}^{F}$ denotes the complexification of ${\mathfrak{n}}$, viewed as a complex linear subspace, a complex Lie subalgebra, of the Lie algebra ${\mathbb{C}}^{F}$ of ${\mathbb{T}}_{{\mathbb{C}}}^{F}$. In view of (4.6), we have, for every $v\in V$, that $N_{{\mathbb{C}}}$ is equal to the set of all $t\in{\mathbb{T}}_{{\mathbb{C}}}^{F}$ such that

[TABLE]

This implies that $N_{{\mathbb{C}}}$ is a closed subgroup of ${\mathbb{T}}_{{\mathbb{C}}}^{F}$ isomorphic to ${\mathbb{T}}_{{\mathbb{C}}}^{F\setminus F_{v}}$, and therefore $N_{{\mathbb{C}}}$ is a reductive complex algebraic group.

If we define

[TABLE]

then it follows from (5.1) that the action of $N_{{\mathbb{C}}}$ on ${\mathbb{C}}^{F}_{v}$ is free and proper. It follows that the action of $N_{{\mathbb{C}}}$ on

[TABLE]

is free and proper, and therefore the $N_{{\mathbb{C}}}$-orbit space

[TABLE]

has a unique structure of a complex analytic manifold of complex dimension $n$ such that the canonical projection from ${\mathbb{C}}^{F}_{\Delta}$ onto $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ exhibits ${\mathbb{C}}^{F}_{\Delta}$ as a principal $N_{{\mathbb{C}}}$-bundle over $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$. On $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ we still have the complex analytic action of the complex Lie group group ${\mathbb{T}}_{{\mathbb{C}}}^{F}/N_{{\mathbb{C}}}$, which is isomorphic to the complexification $T_{{\mathbb{C}}}$ of our real torus $T$ induced by the projection $\pi$. The complex analytic manifold $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ together with the complex analytic action of $T_{{\mathbb{C}}}$ on it is the toric variety defined by the polytope $\Delta$ in the title of this section.

If $v\in V$ and $z\in{\mathbb{C}}^{F}_{v}$, then it follows from (5.1) that there is a unique $t\in N_{{\mathbb{C}}}$ such that $t_{f}=z_{f}$ for every $f\in F\setminus F_{v}$, or in other words, $z=t\cdot\zeta$, where $\zeta\in{\mathbb{C}}^{F}$ is such that $\zeta_{f}=1$ for every $f\in F\setminus F_{v}$. Let $S_{v}:{\mathbb{C}}^{F_{v}}\to{\mathbb{C}}^{F}_{v}$ be defined by $S_{v}(z^{v})_{f}=z^{v}_{f}$ when $f\in F$ and $S_{v}(z^{v})=1$ when $f\in F\setminus F_{v}$, as in Audin [2, p. 159]. If $P_{v}:{\mathbb{C}}^{F}_{v}\to{\mathbb{C}}^{F}_{v}/N_{{\mathbb{C}}}$ denotes the canonical projection from ${\mathbb{C}}^{F}_{v}$ onto the open subset $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{v}:={\mathbb{C}}^{F}_{v}/N_{{\mathbb{C}}}$ of $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$, then $\Psi_{v}:=P_{v}\circ S_{v}$ is a complex analytic diffeomorphism from ${\mathbb{C}}^{F_{v}}$ onto $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{v}$. It is $T_{{\mathbb{C}}}$-equivariant if we let $T_{{\mathbb{C}}}$ act on ${\mathbb{C}}^{F_{v}}$ via ${\mathbb{T}}^{F_{v}}_{{\mathbb{C}}}$ as in Lemma 3.3. We use the diffeomorphism $\Phi_{v}:={\Psi_{v}}^{-1}$ from $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{v}$ onto ${\mathbb{C}}^{F_{v}}$ as a coordinatization of the open subset $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{v}$ of $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$.

If $v,\,w\in V$, then

[TABLE]

Moreover, with a similar argument as for Lemma 4.1, actually much simpler, we have that for every $z^{v}\in U^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{v,\,w}$ the element $z^{w}:=\Phi_{w}\circ{\Phi_{v}}^{-1}(z^{v})\in{\mathbb{C}}^{F_{w}}$ is given by

[TABLE]

In this way the coordinate transformation $\Phi_{w}\circ{\Phi_{v}}^{-1}$ is a Laurent monomial mapping, much simpler than the coordinate transformation (4.3), (4.4). It follows that the toric variety $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ can be alternatively described as obtained by gluing the $n$-dimensional complex vector spaces ${\mathbb{C}}^{F_{v}}$, $v\in V$, 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

$M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$* is simply connected.*

Proof   Let $w\in V$. It follows from (5.5), for all $v\in V$, that the complement of $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{w}$ in $M_{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ is equal to the union of finitely closed complex analytic submanifolds of complex codimension one, whereas $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{w}$ is contractible because it is diffeomorphic to the complex vector space ${\mathbb{C}}^{F_{w}}$. Because complex codimension one is real codimension two, any loop in $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ with base point in $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{w}$ can be slightly deformed to such a loop which avoids the complement of $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{w}$ in $M_{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$, that is, which is contained in $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{w}$, after which it can be contracted within $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{w}$ to the base point in $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{w}$.

$\Box$

Recall the definition in Section 2 of the reduced phase space $M=Z/N$.

Theorem 5.2

The identity mapping from $Z$ into ${\mathbb{C}}^{F}_{\Delta}$, followed by the canonical projection $P$ from ${\mathbb{C}}^{F}_{\Delta}$ to $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}={\mathbb{C}}^{F}_{\Delta}/N_{{\mathbb{C}}}$, induces a $T$-equivariant diffeomorphism $\varpi$ from $M=Z/N$ onto $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$. It follows that each $N_{{\mathbb{C}}}$-orbit in ${\mathbb{C}}^{F}_{\Delta}$ intersects $Z$ in an $N$-orbit in $Z$.

Proof   Because $N$ is a closed Lie subgroup of $N_{{\mathbb{C}}}$, we have that the mapping $P:Z\to{\mathbb{C}}^{F}_{\Delta}/N_{{\mathbb{C}}}$ induces a mapping $\varpi:Z/N\to{\mathbb{C}}^{F}_{\Delta}/N_{{\mathbb{C}}}$, which moreover is smooth.

If $v\in V$, $z\in Z_{v}$, then it follows from (5.1) that the $t_{f}$, $f\in F\setminus F_{v}$, of an element $t\in N_{{\mathbb{C}}}$ can take arbitrary values, and therefore the $|z_{f}|$, $f\in F\setminus F_{v}$ can be moved arbitrarily by means of infinitesimal $N_{{\mathbb{C}}}$-actions. Because $Z$ is defined by prescribing the $|z_{f}|$, $f\in F\setminus F_{v}$, as a smooth function of the $z_{f}$, $f\in F_{v}$, and the $Z_{v}$, $v\in V$, form an open covering of $Z$, this shows that at each point of $Z$ the $N_{{\mathbb{C}}}$-orbit is transversal to $Z$, which implies that $\varpi$ is a submersion.

It follows that $\varpi(M)$ is an open subset of $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$. Because $M$ is compact and $\varpi$ is continuous, $\varpi(M)$ is compact, and therefore a closed subset of $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$. Because $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ is connected, the conclusion is that $\varpi(M)=M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$, that is, $\varpi$ is surjective.

Because $\varpi$ is a surjective submersion, $\!\!\mathop{\rm~{}dim}\nolimits_{{\mathbb{R}}}M=2n=\!\!\mathop{\rm~{}dim}\nolimits_{{\mathbb{R}}}M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$, and $M$ is connected, we conclude that $\varpi$ is a covering map. Because $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ is simply connected, see Lemma 5.1, we conclude that $\varpi$ is injective, that is, $\varpi$ is a diffeomorphism.

$\Box$

**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.

$\oslash$

Note that in the definition of the toric variety $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$, the real numbers $\lambda_{f}$, $f\in F$, did not enter, whereas these numbers certainly enter in the definition of $M$, the symplectic form on $M$, and the diffeomorphism $\varpi:M\to M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$. Therefore the symplectic form $\sigma^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{\lambda}:=(\varpi^{-1})^{*}(\sigma)$ on $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ on $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ will depend on the choice of $\lambda\in{\mathbb{R}}^{F}$. On the symplectic manifold $(M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits},\,\sigma^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{\lambda})$, the action of the maximal compact subgroup $T$ of $T_{{\mathbb{C}}}$ is Hamiltonian, with momentum mapping equal to

[TABLE]

where $\mu^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{\lambda}(M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits})=\Delta$, where we note that $\Delta$ in (2.1) depends on $\lambda$.

In the following lemma we compare the reduced phase space coordinatizations with the toric variety coordinatizations.

Lemma 5.4

Let $v\in V$. Then $M_{v}^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}=\varpi(M_{v})$, and

[TABLE]

is a ${\mathbb{T}}^{F_{v}}$-equivariant diffeomorphism from $U_{v}$ onto ${\mathbb{C}}^{F_{v}}$.

For each $z^{v}\in U_{v}$, the element $\zeta^{v}:=\theta_{v}(z^{v})$ is given in terms of $z^{v}$ by

[TABLE]

where the functions $r_{f^{\prime}}:\Delta\to{\mathbb{R}}_{\geq 0}$ are given by (3.7). We have

[TABLE]

and $z^{v}={\theta_{v}}^{-1}(\zeta_{v})$ is given in terms of $\zeta^{v}$ by

[TABLE]

where $\xi$ is the element of $\Delta$ 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 $z^{v}\in\rho^{v}(Z)$, then $z^{v}\in U_{v}$ if and only if $z^{v}_{f^{\prime}}\neq 0$ for every $f^{\prime}\in F\setminus F_{v}$. That is, the set $Z_{v}$ in Proposition 3.4 is equal to $Z\cap{\mathbb{C}}^{F}_{v}$. It therefore follows from Theorem 5.2 that each $N_{{\mathbb{C}}}$-orbit in the $N_{{\mathbb{C}}}$-invariant subset ${\mathbb{C}}^{F}_{v}$ of ${\mathbb{C}}^{F}_{\Delta}$ intersects the $N$-invariant subset $Z_{v}$ of $Z$ in an $N$-orbit in $Z_{v}$, that is,

[TABLE]

If $z^{v}\in U_{v}$, then Proposition 3.4 implies that $s_{v}(z^{v})_{f}=z^{v}_{f}$ for every $f\in F_{v}$ and

[TABLE]

If we define $t\in{\mathbb{T}}^{F}_{{\mathbb{C}}}$ by

[TABLE]

then $(t\cdot s_{v}(z^{v}))_{t^{\prime}}=1$ for every $t^{\prime}\in F\setminus F_{v}$ and, for every $f\in F_{v}$, $\zeta^{v}_{f}:=(t\cdot s_{v})_{f}$ is equal to the right hand side of (5.9). That is, $t\cdot s_{v}(z^{v})=S_{v}(\zeta^{v})$, see the definition of $S_{v}$ in the paragraph preceding (5.5). On the other hand, it follows from (5.1) that $t\in N_{{\mathbb{C}}}$, and therefore

[TABLE]

that is, $\zeta^{v}={\Psi_{v}}^{-1}\circ\varpi\circ\psi_{v}(z^{v})$.

$\Box$

Corollary 5.5

Let $s$ be the relative interior of a face of $\Delta$. Then ${\mu_{T}}^{-1}(s)$ is equal to a stratum $S$ of the orbit type stratification in $M$ of the $T$-action, and also equal to the preimage under $\varpi:M\to M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ of a $T_{{\mathbb{C}}}$-orbit in $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$. If $s=\{v\}$ for a vertex $v$, then ${\mu_{T}}^{-1}(s)=\{m_{v}\}$ for the unique fixed point $m_{v}$ in $M$ for the $T$-action such that $\mu_{T}(m_{v})=v$.

The mapping $s\mapsto{\mu_{T}}^{-1}(s)$ is a bijection from the set $\Sigma_{\Delta}$ of all relative interiors of faces of $\Delta$ onto the set $\Sigma$ of all strata of the orbit type stratificiation in $M$ for the action of $T$. If $s,\,s^{\prime}\in\Sigma_{\Delta}$ then $s$ is contained in the closure of $s^{\prime}$ in $\Delta$ if and only if ${\mu_{T}}^{-1}(s)$ is contained in the closure of ${\mu_{T}}^{-1}(s^{\prime})$ in $M$.

The domain of definition $M_{v}$ of $\varphi_{v}$ in $M$ is equal to the union of the $S\in\Sigma$ such that $m_{v}$ belongs to the closure of $S$ in $M$. The domain of definition $M_{v}^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}=\varpi(M_{v})$ of $\Phi_{v}$ is equal to the union of the corresponding strata of the $T$-action in $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$, each of which is a $T_{{\mathbb{C}}}$-orbit in $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$. $M_{v}$ and $M_{v}^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ are open cells in $M$ and $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$, respectively.

Proof   There exists a vertex $v$ of $\Delta$ such that $v$ belongs to the closure of $s$ in ${\mathfrak{t}}^{*}$, which implies that $s$ is disjoint from all $f^{\prime}\in F\setminus F_{v}$. Let $F_{v,\,s}$ denote the set of all $f\in F_{v}$ such that $s\subset f$, where $F_{v,\,s}=\emptyset$ if and only if $s$ is the interior of $\Delta$. For any subset $G$ of $F_{v}$, let ${\mathbb{C}}^{F_{v}}_{G}$ denote the set of all $z\in{\mathbb{C}}^{F_{v}}$ such that $z_{f}=0$ if $f\in G$ and $z_{f}\neq 0$ if $f\in F_{v}\setminus G$. It follows from $\mu_{v}=\mu_{T}\circ\psi_{v}$ and (3.8) that $\psi_{v}^{-1}({\mu_{T}}^{-1}(s))$ is equal to $U_{v}\cap{\mathbb{C}}^{F_{v}}_{G}$ with $G=F_{v,\,s}$. The diffeomorphism $\theta_{v}$ maps this set onto the set ${\mathbb{C}}^{F_{v}}_{G}$ with $G=F_{v,\,s}$. Because the sets of the form ${\mathbb{C}}^{F_{v}}_{G}$ with $G\subset F_{v}$ are the strata of the orbit type stratification of the ${\mathbb{T}}^{F_{v}}$-action on ${\mathbb{C}}^{F_{v}}$, and also equal to the $({\mathbb{T}}_{{\mathbb{C}}})^{F_{v}}$-orbits in ${\mathbb{C}}^{F_{v}}$, the first statement of the corollary follows.

The second statement follows from ${\mu_{v}}^{-1}(\{v\})=\{0\}$ and the fact that [math] is the unique fixed point of the ${\mathbb{T}}^{F_{v}}$-action in $U_{v}$.

If $s\in\Sigma_{\Delta}$ and $v\in V$, then $m_{v}$ belongs to the closure of ${\mu_{T}}^{-1}(s)$ if and only if $s$ is not contained in any $f^{\prime}\in F\setminus F_{v}$. This proves the characterization of the domain of definition $M_{v}:=Z_{v}/N={\mu_{T}}^{-1}(\Delta_{v})$ of $\varphi_{v}$. The last statement follows from the fact that $\Phi_{v}$ is a diffeomorphism from $M_{v}^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ onto the vector space ${\mathbb{C}}^{F_{v}}$, and $\varpi$ is a diffeomorphism from $M_{v}$ onto $M_{v}^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$.

$\Box$

**Remark 5.6  ** If $v,\,w\in V$, then

[TABLE]

Using the formula (5.6) for $\Phi_{w}\circ{\Phi_{v}}^{-1}$, 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 $\xi:=\mu_{v}(z^{v})=\mu_{w}(z^{w})$, $|z^{v}_{f}|=r_{f}(\xi)$ if $f\in F_{v}\setminus F_{w}$, and

[TABLE]

if $f^{\prime}\in F\setminus F_{v}$ and $g\in F_{w}$.

$\oslash$

In the following corollary we describe the symplectic form $\sigma^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{\lambda}$ on the toric variety $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ in the toric variety coordinates.

Corollary 5.7

For each $v\in V$, the symplectic form $({\Phi_{v}}^{-1})^{*}(\sigma^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}_{\lambda})$ on ${\mathbb{C}}^{F_{v}}$ is equal to $({\theta_{v}}^{-1})^{*}(\sigma_{v})$, where $\sigma_{v}$ is the standard symplectic form on ${\mathbb{C}}^{F_{v}}$ given by (3.4).

Because $r_{f^{\prime}}(\mu_{v}(z^{v}))^{2}$ is an inhomogeneous linear function of the quantities $|z_{f}^{v}|^{2}$, it follows from (5.9) that the equations which determine the $|z^{v}_{f}|^{2}$ in terms of the quantities $|\zeta_{f}^{v}|^{2}$ are $n$ polyomial equations for the $n$ unknowns $|z^{f}_{v}|^{2}$, $f\in F_{v}$, where the coefficients of the polyomials are inhomogeneous linear functions of the $|\zeta_{f}^{v}|$, $f\in F_{v}$. In this sense the $|z_{f}^{v}|^{2}$, $f\in F_{v}$, are algebraic functions of the $|\zeta^{v}_{f}|^{2}$, $f\in F_{v}$, and substituting these in (5.9) we obtain that the diffeomorphism ${\theta_{v}}^{-1}$ from ${\mathbb{C}}^{F_{v}}$ onto $U_{v}$ is an algebraic mapping. If $\Delta$ is a simplex, when $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ is the $n$-dimensional complex projective space, we have an explicit formula for ${\theta_{v}}^{-1}$, see Subsection 6.1. However, already in the case that $\Delta$ is a planar quadrangle, when $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ is a complex two-dimensional Hirzebruch surface, we do not have an explicit formula for ${\theta_{v}}^{-1}$. 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 $\lambda$-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 $\lambda$, 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 $\Delta$, where in general we do not have a really explicit formula for the momentum mapping in toric variety coordinates.

$\oslash$

6 Examples

6.1 The complex projective space

Let $\Delta$ be an $n$-dimensional simplex in ${\mathfrak{t}}^{*}$. A little bit of puzzling shows that there is a ${\mathbb{Z}}$-basis $e_{i}$, $1\leq i\leq n$, of the integral lattice ${\mathfrak{t}}_{{\mathbb{Z}}}$ in ${\mathfrak{t}}$, such that, with the notation

[TABLE]

the $X_{f}$, $f\in F$, are the $e_{i}$, $0\leq i\leq n$. That is, in the sequel we write $F=\{0,\,1,\,\ldots,\,n\}$. The Delzant simplex (2.1) is determined by the inequalities $\langle e_{i},\,\xi\rangle+\lambda_{i}\geq 0$, $0\leq i\leq n$, which has a non-empty interior if and only if

[TABLE]

In the sequel we take for $v$ the vertex determined by the equations $\langle e_{i},\,\xi\rangle+\lambda_{i}=0$ for all $1\leq i\leq n$, where $F_{v}=\{1,\,\ldots,\,n\}$. If we write $\xi_{i}=\langle e_{i},\,\xi\rangle$, $1\leq i\leq n$, when $\xi\in{\mathfrak{t}}^{*}$, then (3.2) yields that

[TABLE]

It follows from (3.7) that

[TABLE]

and therefore (5.9) yields that

[TABLE]

where we have written

[TABLE]

Note that $U_{v}$ is the open ball in ${\mathbb{C}}^{n}$ with center at the origin and radius equal to $(2c)^{1/2}$.

The equations (6.3) imply that

[TABLE]

hence

[TABLE]

Therefore the mapping ${\theta^{v}}^{-1}:\zeta^{v}\mapsto z^{v}$ is given by the explicit formulas

[TABLE]

It can be verified that the symplectic form $({\theta_{v}}^{-1})^{*}(\sigma_{v})$, where

[TABLE]

is the standard symplectic form in (3.4), is equal to $\gamma$ times the Fubini-Study form in Griffiths and Harris [6, p. 30, 31]. In view of Remark 3.7 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 $\Delta$ be a quadrangle in the ${\mathfrak{t}}^{*}$ plane. A little bit of puzzling shows that there is an $m\in{\mathbb{Z}}_{\geq 0}$ and a ${\mathbb{Z}}$-basis $e_{1}$, $e_{2}$ of the integral lattice ${\mathfrak{t}}_{{\mathbb{Z}}}$ in ${\mathfrak{t}}$, such that the $X_{f}$, $f\in F$, are the $e_{i}$, $1\leq i\leq 4$, with $e_{3}=\,-e_{1}+m\,e_{2}$, and $e_{4}=\,-e_{2}$. We recognize the toric variety $M^{\!\!\mathop{\mbox{\rm\scriptsize~{}toric}}\nolimits}$ as the Hirzebruch surface $\Sigma_{m}$, see Hirzebruch [9].

The Delzant polytope (2.1) is determined by the inequalities $\langle e_{i},\,\xi\rangle+\lambda_{i}\geq 0$, $1\leq i\leq 4$, which is a quadrangle if and only if

[TABLE]

which inequalities imply that $\lambda_{2}+\lambda_{4}=\gamma_{+}+\gamma_{-}>0$.

In the sequel we take for $v$ the vertex determined by the equations $\langle e_{i},\,\xi\rangle+\lambda_{i}=0$ for $i=1,\,2$, where $F_{v}=\{1,\,2\}$. If we write $\xi_{i}=\langle e_{i},\,\xi\rangle$, $1\leq i\leq 2$, when $\xi\in{\mathfrak{t}}^{*}$, then (3.2) yields that

[TABLE]

It follows from (3.7) that

[TABLE]

and therefore (5.9) yields that

[TABLE]

If we write $t_{i}=|z^{v}_{i}|^{2}$ and $\tau_{i}=|\zeta^{v}_{i}|^{2}$, then this leads to the equations

[TABLE]

for $t_{1},\,t_{2}$. If we solve $t_{1}$ from the first equation,

[TABLE]

and substitute this into the second equation, then this leads to the polynomial equation

[TABLE]

of degree $m+1$ for $t_{2}$. If we substract the left hand side from the right hand side then the derivative with respect to $t_{2}$ is strictly positive, and one readily obtains that for every $\tau_{1},\,\tau_{2}\in{\mathbb{R}}_{\geq 0}$ there is a unique solution $t_{2}\in{\mathbb{R}}_{\geq 0}$, confirming the first statement in Lemma 5.4.

On the other hand, if we work over ${\mathbb{C}}$, and view both the parameter $\varepsilon:=(1+\tau_{1})^{m}\,\tau_{2}$ and the unknown $t_{2}$ as elements of the complex projective line ${\mathbb{P}}^{1}$, then the equation (6.8) defines a complex algebraic curve $C$ in the $(t_{2},\,\varepsilon)$-plane ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$, where the restriction to $C$ of the projection to the first variable $t_{2}$ is a complex analytic diffeomorphism from $C$ onto ${\mathbb{P}}^{1}$, as on $C$ we have that $\varepsilon$ is a complex analytic function of $t_{2}$. In particular $C$ is irreducible. The restriction to $C$ of the projection to the second variable $\varepsilon$ is an $(m+1)$-fold branched covering. Over $\varepsilon=0$ and over $\varepsilon=\infty$ we have that $m$ of the $m+1$ branches come together, whereas there are two more branch points on the $\varepsilon$-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 $t_{2}$ of (6.8), as a complex analytic function of $\varepsilon$ in the complement of the branch points, will reach each other branch if $\varepsilon$ runs over a suitable loop. In other words, the solution $t_{2}$ is an algebraic function of $\varepsilon$ of degree $m+1$, and no branch of a solution is of lower degree. This holds in particular for our solutions $t_{2}\in{\mathbb{R}}_{\geq 0}$ for $\varepsilon\in{\mathbb{R}}_{\geq 0}$.