Reduced Phase Space and Toric Variety Coordinatizations Of

8/2/2019 Reduced Phase Space and Toric Variety Coordinatizations Of

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Under consideration for publication in Math. Proc. Camb. Phil. Soc. 147

Reduced phase space and toric variety coordinatizations of

Delzant spaces

By JOHANNES J. DUISTERMAAT AND ALVARO PELAYO

Mathematisch Instituut, Universiteit UtrechtP.O. Box 80 010, 3508 TA Utrecht, The Netherlands.

e-mail: [email protected]

and

University of CaliforniaBerkeley, Mathematics Department,970 Evans Hall # 3840

Berkeley, CA 94720-3840, USA.e-mail: [email protected]

(Received 6 May 2008; revised 11 August 2008)

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 geometryview-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, ) be a smooth compact and connected symplectic manifold of dimension 2n

and let T be a torus which acts effectively on (M, ) by means of symplectomorphisms.

If the action of T on (M, ) is moreover Hamiltonian, then dim T n, and the imageof the momentum mapping T : 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 dim T = n, (M, ) 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

Research stimulated by KNAW professorship Research partially funded by an NSF postdoctoral fellowship


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148 Johannes J. Duistermaat and Alvaro Pelayo

of t. Thirdly Delzant [3, pp. 328, 329] proved that for every Delzant polytope there

exists a Delzant space such that T(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 N of the momentum mapping of the Hamiltonian N-action, where

E, N and N are determined by the Delzant polytope.

Finally Delzant [3, Sec. 5] observed that the Delzant polytope gives rise to a fan

(= eventail in French), and that the Delzant space with Delzant polytope is T-

equivariantly diffeomorphic to the toric varietyMtoric defined by the fan. Here Mtoric 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

TC of T. In our description in Section 5 of the toric variety Mtoric 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

v of a T-invariant open cell Mv in M, where Mv contains the unique fixed point mvin M of the T-action such that T(mv) = v. We give an explicit construction of the

inverse v of v, which is a maximal diffeomorphism in the sense of Remark 39. The

construction of v originated in an attempt to extend the equivariant symplectic ball

embeddings from (B2nr , 0) (Cn, 0) into the Delzant space (M, ) in Pelayo [11] by

maximal equivariant symplectomorphisms from open neighborhoods of the origin in Cn

into the Delzant space (M, ). Ifv and w are two different vertices, then the coordinate

transformation w v1 is given by the explicit formulas (4.3), (4.4). This system of

coordinates gives a new construction of the symplectic manifold with torus action from

the Delzant polytope. After we wrote this paper V. Guillemin informed us that he had

also considered the idea of this construction.Let be the set of all strata of the orbit type stratification of M for the T-action. Then

the domain of definition Mv of v is equal to the union of all S such that the fixed

point mv belongs to the closure of S in M, see Corollary 56. The strata S 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 v is equal to the domain of definition

of a natural complex analytic TC-equivariant coordinatization v of a TC-invariant open

cell. The diffeomorphism v v1, which sends the reduced phase space coordinates

to the toric variety coordinates, maps Uv := v(Mv) diffeomorphically onto a complex

vector space, and is given by the explicit formulas (5.10).

In the toric variety coordinates the complex structure is the standard one and the coor-

dinate transformations are relatively simple Laurent monomial transformations, whereas

the symplectic form is generally given by quite complicated algebraic functions. On theother hand, in the reduced phase space coordinates the symplectic form is the stan-

dard one, but the coordinate transformations, and also the complex structure, have a

more complicated appearance. While these completely explicit coordinate formulas are

the main novelty of the paper, we also use them to reprove many of the known results,

leading to an efficient and hopefully attractive exposition of the subject.

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


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Coordinatizations of Delzant spaces 149

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 v and v take their values in

CFv . This leads to a natural projection v : CF 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 tZappears, 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 [8] 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 discuss the de Rham

cohomology classes of Kahler forms on the toric manifold, which actually are equal to

the de Rham cohomology classes of the symplectic forms of the model Delzant spaces.

In Section 7 we present these computations for the two simplest classes of examples, thecomplex 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 Rn/Zn. For this reason, tZ is called

the integral lattice in t. However, because we do not have a preferred Z-basis oftZ, we do

not write T = Rn/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 [12, p.162], every face of is compact. For every v V, we


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write

Fv = {f F | v f}. is called a Delzant polytope if it has the following properties, see Guillemin [8, p. 8].

i) For each f F there is an Xf tZ and f R such that the hyperplane which

contains f is equal to the set of all t such that Xf, + f = 0, and is

contained in the set of all t such that Xf, + f 0. The vector Xf and

constant f 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

= { t | Xf, + f 0 for every f F}. (2.1)

Also, #(Fv) = n for every v V, which already makes the polytope quite special. Inthe 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 be the real linear map from RF to t defined by

(t) :=fF

tf Xf, t RF. (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 (ZF) = tZ and (RF) = t. It follows that induces a surjective

homomorphism of Lie groups from the torus RF/ZF = (R/Z)F onto t/tZ, and we have

the corresponding surjective homomorphism exp from RF/ZF onto T.

Write n := ker , a linear subspace ofRF, and N = ker(exp ), a compact commuta-

tive subgroup of the torus RF/ZF. Actually, N is connected, see Lemma 31 below, andtherefore 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 RF/ZF, where t RF/ZF maps z CF to the element t z CF

defined by

(t z)f = e2 i tf zf, f F.

The infinitesimal action of Y RF = Lie(RF/ZF) is given by

(Y z)f = 2 i Yf zf,

which is a Hamiltonian vector field defined by the function

z Y, (z) = fF

Yf |zf|2/2 =

fF

Yf (xf2 + yf

2)/2, (2.3)

and with respect to the symplectic form

:= (i /4)fF

dzf dzf = (1/2)fF

dxf dyf, (2.4)

if zf = xf + i yf, with xf, yf R. Here the factor 1/2 is introduced in order to avoid

an integral lattice (2 Z)F instead of our ZF.

1 We did not find a proof of the connectedness of N in [3], [2], or [8].


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Because the right hand side of (2.3) depends linearly on Y, we can view (z) as an

element of (RF) RF, with the coordinates

(z)f = |zf|2/2 = (xf

2 + yf2)/2, f F. (2.5)

In other words, the action ofRF/ZF on CF is Hamiltonian, with respect to the symplectic

form and with momentum mapping : CF (Lie(RF/ZF)) given by (2.3), or

equivalently (2.5).

It follows that the subtorus N ofRF/ZF acts on CF in a Hamiltonian fashion, with

momentum mapping

N := n

: CF n, (2.6)

where n : n RF denotes the identity viewed as a linear mapping from n RF to RF,

and its transposed n

: (RF) n is the map which assigns to each linear form on RF

its restriction to n.

Write N = n

(), where denotes the element of (RF) RF with the coordinates

f, f F. It follows from Guillemin [8, Th. 1.6 and Th. 1.4] that N is a regular value of

N, hence the level set Z := N1({N}) of N for the level N is a smooth submanifold

ofCF, 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 M on M such that pM = Z

, where Z is the identity viewed as a

smooth mapping from Z to CF.

Remark 21 Guillemin [8] used the momentum mapping N N instead ofN, 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 N for the N-action, which

does not depend on , and do the reduction at the level N.

The symplectic manifold (M, M) is the Marsden-Weinstein reductionof the symplec-

tic manifold (CF, ) for the Hamiltonian N-action at the level 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 (RF/ZF)/N T, with momentum mapping T : M t

determined by

T p = ( )|Z . (2.7)

The torus T acts effectively on M and T(M) = , see Guillemin [8, 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 defined by , see Guillemin, [8, p. 13]. This proves the existence part [3,

pp. 328, 329] of Delzants theory.

3. The reduced phase space coordinatizations.

For any v V, let v := v : (RFv) (RF) denote the transposed of the restriction

projection v : RF RFv . If in the usual way we identify (RFv) and (RF) with RFv

and RF, respectively, then v : RFv RF is the embedding defined by v(x)f = xf

if f Fv and v(x)f = 0 i f f F, f / Fv. Because v maps ZFv into ZF and


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v(RFv) ZF = v(Z

Fv), it induces an embedding of the n-dimensional torus RFv/ZFv

into RF/ZF, which we also denote by v.

Lemma 31. With these notations, RF, ZF, andRF/ZF are the direct sum of n and

v(RFv), n Zn and v(ZFv), and N and v(RFv/ZFv), respectively.

It follows that N is connected, a torus, with integral lattice equal to n ZF. It also

follows that v 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 oft, there exists a unique

tv RFv , such that

(t) =fFv

(tv)f Xf = (v(tv)),

that is, t v(tv) n. Moreover, because the Xf, f Fv, also form a Z-basis of tZ, we

have that tv ZFv , and therefore t v(tv) n ZF, if t ZF.

Lemma 32. We have z Z if and only if (z) (t). More explicitly, if and

only if there exists a t such that

|zf|2/2 f = Xf, for every f F. (3.1)

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

Furthermore, Z = ( )1(()), ( )(Z) = (), and Z is a compact subset

ofCF.

Proof. The kernel of n

is equal to the space of all linear forms on RF which vanish

on n := ker , and therefore ker n

is equal to the image of : t (RF). Because is

surjective, is injective, which proves the uniqueness of .

It follows from (3.1) that Xf, + f 0 for every f F, and therefore inview of (2.1). Conversely, if , then there exists for every f F a complex number

zf such that |zf|2/2 = Xf, +f, which means that z Z and ()(z) =

(). The

set () is compact because is compact and is continuous. Because the mapping

is proper, it follows that Z = ( )1(()) is compact.

Let v V. The Xf, f Fv, form an R-basis of t, and therefore there exists for each

z CFv a unique = v(z) t such that (3.1) holds for every f Fv. That is, the

mapping v : CFv t is defined by the equations

|zf|2/2 f = Xf, v(z), z C

Fv , f Fv. (3.2)

In other words, v is defined by the formula

v v = v ( ) v, (3.3)

where v denotes the restriction projection from RF onto RFv .

Lemma 33. If we let T act onCFv viaRFv/ZFv by means of (t, z) ( v)1(t) z,

then v : CFv t is a momentum mapping for this Hamiltonian action of T onCFv ,

with v(0) = v. Here the symplectic form onCFv is equal to

:= (i /4)fFv

dzf dzf = (1/2)fFv

dxf dyf, (3.4)

that is, (2.4) with F replaced by Fv.


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Letv denote the restriction projection fromCF onto CFv , and let Uv be the interior

of the subset v(Z) ofCFv . Write

v := \

fF\Fv

f. (3.5)

Then v(Z) = v1(), v(v(Z)) = , Uv = v

1(v), and v(Uv) = v. In particu-

lar v(Z) is a compact subset ofCFv , and Uv is a bounded and connected open neighbor-

hood of 0 inCFv .

Proof. The first statement follows from (3.3), the fact that v v is a momen-

tum mapping for the standard RFv/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, v(0)+f = 0 for every f Fv,

hence 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 Fv.It follows from (3.2), Lemma 32, that z Z if and only if

|zf|2/2 = Xf, v(v(z)) + f 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 v((z)) . Conversely, if , then it

follows from Lemma 32 that there exists z Z such that () = (z) , of which the

restriction to Fv yields = v((z)).

If v, zv CFv , v(z

v) = , then Xf, v(zv) + f > 0 for every f

F \ Fv,

which will remain valid if we replace zv by zv in a sufficiently small neighborhood ofzvin CFv . It follows that we can find

z CF such that v(

z) =

zv and (3.6) holds with z

replaced by z. That is, z Z, and we have proved that zv Uv.

Let conversely z Uv CFv . We have in view of (3.2) that

|zf|2/2 = Xf, v(z) v(0) = Xf, v(z) v

for every f Fv. Therefore v(z) v is multiplied by c2 if we replace z by c z, c > 0.

Because z is in the interior ofv(Z), we have c zv v(Z), hence v(c z) for c > 1, c

sufficiently close to 1. On the other hand, if belongs to a face of which is not adjacent

to v, then v + ( v) / for any > 1. It follows that v(z) does not belong to any

f F \ Fv, that is, v(z) v.

The equation (3.6) can be written in the form |zf| = rf(v(v(z))), where, for each

f F, the function rf : R0 is defined by

rf() := (2(Xf, + f))1/2, f F, . (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 zv = v(z).

Ifzv Uv, then for each f F\ Fv the coordinate zf lies on the circle about the origin

with strictly positive radius rf(v(zv). Lemma 31 implies that the homomorphism which

assigns to each element of N its projection to RF\Fv/ZF\Fv is an isomorphism, and the

latter torus is the group of the coordinatewise rotations of the zf, f F \ Fv.

On the other hand, if we let Zv := v1(Uv) Z, which is an open subset of Z C

F,

the differential of N in (2.6) evaluated at any z = (zf)fF Zv is surjective; indeed,

write zf = xf + i yf for every f F and suppose by contradiction that N is not


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surjective. Then there exists X n, X = 0, such that

fF Xf (xf xf + yf y

f) = 0 for

every z, z CF. Because z Zv, zf = 0 for every f F \ Fv. By taking f

F \ Fv,

and zf = xf + i y

f = 0 for every f = f

, as well as zf arbitrary, we conclude that

Xf = 0. Because f is arbitrary, X v(RFv). On the other hand by Lemma 31,

n v(RFv) = {0}, so X = 0, a contradiction.

These facts lead us to the following conclusions, where in order to make the presentation

self-contained, we do not assume that Z is a smooth submanifold ofCF of codimension

equal to the dimension of N.

Proposition 34. Letv be a vertex of. The open subsetZv := v1(Uv) Z ofZ is

a connected smooth submanifold ofCF of real dimension2n+(dn), where d = #(F) and

d n = dim N. The action of the torus N on Zv is free, and the projection v : Zv Uvexhibits 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 v : Zv Uv such thatsv(z)f R>0for every z Uv and f

F \ Fv. Actually, sv(z)f = rf(v(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 v := pv sv is a T-equivariant symplectomorphism from

Uv onto Mv, where T acts on Uv viaRFv/ZFv , as in Lemma 33.

Remark 35 When z belongs to the closure v(Z) = Uv of Uv in CFv , see Lemma

33, we can define sv(z) CF by sv(z)f = zf when f Fv and sv(z)f = rf(v(z))

when f F \ Fv. This defines a continuous extension sv : Uv CF of the mapping

sv : Uv Z. Therefore sv(Uv) Z, and v := p sv : Uv M is a continuous extension

of the diffeomorphism v : Uv Mv.

The continuous mapping v : Uv M is surjective, but the restriction of it to the

boundary Uv := Uv \ Uv ofUv in CFv is not injective. If zv Uv, then the set G of all

f F \ Fv such that sv(zv)f = 0, or equivalently T(v(z

v)) f, is not empty. The

fiber of v over v(zv) is equal to the set of all tv zv, where the tv RFv/ZFv are of

the form

tvf = gG

(v)fg tg, f Fv,

where tg R/Z. It follows that each fiber is an orbit of some subtorus ofRFv/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 36. The sets Zv, v V, form a covering of Z. As a consequence, we

recover the results mentioned in Section 2 that Z is a smooth submanifold ofCF 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 mappingT :

M T as in (2.7). Since Z is the level set of N for the level N, it follows thatN is

a regular value of N.


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Moreover, the sets Mv, v V, form an open covering of M and the v := (v)1 :

Mv Uv form an atlas ofT-equivariant symplectic coordinatizations of the Hamiltonian

T-space M. For each v V, we have Mv = T1(v), and T|Mv = v v.

For a characterization of Mv in terms of the orbit type stratification in M for the T-

action, see Corollary 56, which also implies that Mv is an open cell in M.

Corollary 37. For every f F the set T1(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 v : Uv Mv is

maximal among all diffeomorphisms from open subsets ofCFv onto open subsets of M.

Proof. If f F, then for each v V we have that

v1(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 thebeginning of Section 2. On the other hand, v

1(f) = if f / v. Because T1(f)

Mv = v(v1(f)), and the Mv, v V, form an open covering ofM, 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 T1(f) with f F \ Fv.

Remark 38 It follows from the proof of Corollary 37, that T1(f) is a connected

component of the fixed point set in M of the of the circle subgroup exp(RXf) of T.

Actually, T1(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 embed-

ding (t/(RXf)) t is equal to a translate of f.

In a similar way, if g is a k-dimensional face of , then T1(g) is a 2k-dimensional

Delzant space for the quotient of T by the subtorus ofT which acts trivially on T1(g).

Remark 39 Let : T Rn/Zn be an isomorphism of tori, which allows us to let

t T act on Cn via Rn/Zn by means of

(t z)j = e2 i (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 : U M be a T-equivariant

symplectomorphism from U onto an open subset (U) of M. Because 0 is the uniquefixed point for the T-action in U, and the fixed points for the T-action in M are the pre-

images under T of the vertices of , there is a unique v V such that T((0)) = v.

Let Iv : CFv Cn denote the complex linear extension of the tangent map of the torus

isomorphism ( v). In terms of the notation of Lemma 33 and Proposition 34,

we have that U Iv(Uv) and v = Iv on Iv1(U), which leads to an identification

of with the restriction of v to the connected open subset Iv1(U) of Uv, via the

isomorphism Iv1.

The s, with U equal to a ball in Cn centered at the origin, are the equivariant sym-

plectic ball embeddings in Pelayo [11], and the second statement in Corollary 37 shows


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that the diffeomorphisms v are the maximal extensions of these equivariant symplectic

ball embeddings.

4. The coordinate transformations

Recall the description in Lemma 33, for every vertex v, of the open subset Uv =

1v (v) ofCFv .

Let v, w V. Then

Uv, w := v(Mv Mw) = Uv v1 w(Uw)

= {zv Uv | (zv)f = 0 for every f Fv \ Fw}. (4.1)

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

w v1 = w

1 v : 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 (w)gf, g Fw, such that

Xf =gFw

(w)gf Xg. (4.2)

Note that iff Fw, then (w)gf = 1 when g = f and (w)

gf = 0 otherwise. For the following

lemma recall that rg is defined by expression (3.7).

Lemma 41. Let v, w V, zv Uv, w. Then zw := w v

1(zv) Uw CFw is

given by

zwg = fFv

(zvf)(w)gf /

fFv\Fw

|zvf|(w)gf (4.3)

if g Fw Fv, and

zwg =

fFv

(zvf)(w)g

f rg(v(zv))/

fFv\Fw

|zvf|(w)g

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,

sw(zw)f = e

i tf sv(zv)f for every f F

for some t RF such that fF

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 =

fF\Fw

(w)gf tf for every g Fw. (4.6)

Note that v(zv) = T(m) = w(z

w), where m = v(zv) = w(z

w). It follows from

the definition of the sections sv and sw, see Proposition 34, that


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i) sv(zv)f = z

vf and sw(z

w)f = zwf if f Fv Fw,

ii) sv(zv)f = z

vf and sw(z

w)f = rf(w(zw)) = rf(v(z

v)) if f Fv \ Fw,

iii) sv(zv)f = rf(v(zv)) and sw(zw)f = zwf if f Fw \ Fv, and

iv) sv(zv)f = rf(v(z

v)) = rf(w(zw)) = sw(z

w)f if f F \ (Fv Fw).

It follows from ii) and iv) that tf = arg zvf and tf = 0 modulo 2 if f Fv \ Fw and

f F \ (Fv Fw), respectively. Then (4.6) implies that, modulo 2,

tg =

fFv\Fw

(w)gf arg zvf for every g Fw.

It now follows from i) and iii) that if g Fw, then zwg = sw(z

w)g = ei tg sv(z

v)g is

equal to

ei tg zvg =

fFv

(zvf)(w)g

f /

fFv\Fw

|zvf|(w)g

f

if g Fv, and equal to

ei tg |zvg | =

fFv

(zvf)(w)g

f |zvg |/

fFv\Fw

|zvf|(w)g

f

if g / Fv, respectively. Here we have used that if g Fw, then (w)gf = 1 if f = g and

(w)gf = 0 if f Fw, f = g. Because |zvg | = rg(v(z

v)) if g / Fv, see (3.6) and (3.7), this

completes the proof of the lemma.

Remark 42 Note that zv Uv, w means that zv Uv and zvf = 0 if f Fv \ Fw.

Furthermore, zv Uv implies that if g / Fv, then v(zv) / g, and therefore rg is

smooth on a neighborhood of v(zv). Finally, note that if g Fw and f Fv Fw, then

(w)g

f {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 43 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 44. Let, for each v V, the mapping v : CFv t be defined by

(3.2), which is a momentum mapping for a Hamiltonian T-action viaRFv/ZFv on the

symplectic vector space CFv as in Lemma 33. Define Uv := v1(v). If also w V,

define Uv, w as the right hand side of (4.1), and, if zv Uv, w, define w, v(z

v) := zw,

where zw CFw is given by (4.3) and (4.4).

Thenw, v is a T-equivariant symplectomorphism fromUv, w onto Uw, v such thatw =

v v, w on Uw, v. The w, v satisfy the cocycle condition w, v v, u = w, u where the

left hand side is defined. Glueing together the Hamiltonian T-spaces Uv, v V, with the

momentum maps v, by means of the gluing maps w, v, v, w V, we obtain a compact


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connected smooth symplectic manifold

M with an effective Hamiltonian T-action with

a common momentum map : M T such that (M) = . In other words, M is aDelzant space for the Delzant polytope .The Delzant space M is obviously isomorphic to the Delzant space M = 1({})/Nintroduced in Section 2, and actually the isomorphism is used in the proof that M is aDelzant 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 v, 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 u where uf = e2 i tf for every f F is an isomorphism from the torus RF/ZF

ontoT

F

, whereT

F

acts onC

F

by means of coordinatewise multiplication andR

F

/Z

F

acted on CF via the isomorphism from RF/ZF onto TF. The complexification TC of the

compact Lie group T is the multiplicative group C of all nonzero complex numbers, and

the complexification ofTF is equal to TFC

:= (TC)F = (C)F, which also acts on CF by

means of coordinatewise multiplication.

The complexification NC ofN is the subgroup exp(nC) ofUFC

, where nC := ni n CF

denotes the complexification of n, viewed as a complex linear subspace, a complex Lie

subalgebra, of the Lie algebra CF ofTFC

. In view of (4.6), we have, for every v V, that

NC is equal to the set of all t TFC such that

tg =

fF\Fv

tf(v)g

f , g Fv. (5.1)

This implies that NC is a closed subgroup ofTFC isomorphic to TF\FvC , and therefore NCis a reductive complex algebraic group.

If we define

CFv := {z C

F | zf = 0 for every f F \ Fv}, (5.2)

then it follows from (5.1) that the action of NC on CFv is free and proper. It follows that

the action of NC on

CFV =

vV

CFv (5.3)

is free.

Lemma 51. The action of NC onCFV is proper.

Proof. As the referee observed, this does not follow immediately from the properness of

the NC-action on each of the CFv s, and because we did not find a proof in the literature,

we present one here. Also, G. Schwarz observed that in view of Lunas slice theorem

it is sufficient to prove that the NC-orbits are closed subsets of CFV but we could not

find a proof for the closedness of the orbits which is much simpler than the proof of the

properness of the action. Finally, the statement of the lemma is implicitly contained in

the statement in Audin [2, bottom of p. 155] that U X is a principal KC-bundle.

For each subset E of F, let C(E) =

fE R0 Xf denote the polyhedral cone in t

spanned by the vectors Xf, f E. Let v, w V, X C(Fv) C(Fw), and . That


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is,

X = fFv

cf Xf = gFw

dg Xg,

with cf, dg R0, and Xf, + f 0 for every f F. Because Xf, v + f = 0 for

every f Fv, it follows that X, v 0, with equality if and only if cf = 0 or f

for every f Fv.

Similarly X, w 0, with equality if and only if dg = 0 or g for every g Fw.

For = (1/2) (v + w) we have f Fv Fw if f Fv or g Fw, hence

cf = 0 for every f Fv \ Fw and dg = 0 for every g Fw \ Fv. We therefore have proved

that the collection of simplicial cones C(Fv), v V, has the fan property of Demazure

[4, Def. 1 in 4] that C(Fv) C(Fw) = C(Fv Fw) for every v, w V.

The argument below that the fan property implies the properness of the NC-action on

C

FV is inspired by the proof of Danilov [5, bottom of p. 133] that a toric variety defined

by a fan is separated.

What we have to prove is that ifx CFV is close to x0 CFV, t NC, and y = t x C

FV

is close to y0 CFV, then t remains in a compact subset ofNC, that is, tf remains bounded

and bounded away from 0 for every f F. It follows from (5.3) that we have x0 CFvand y0 CFw for some v, w V. Then (5.2) implies that for every f F \ (Fv Fw)

both xf and yf remain bounded and bounded away from zero, hence tf = yf/xf remains

bounded and bounded away from zero. The fan property implies that there exists a linear

form on tsuch that Xf, > 0 for every f Fv\Fw, Xf, = 0 for every f FvFw,

and Xg, < 0 for every g Fw \ Fv. We can arrange that Xf, Z for every f Fv,

which implies that Xf, Z for every f F because the Xf, f Fv form a Z-basis

of F.

For each f Fv \ Fw it follows from (5.1) that

tf =

gFw\Fv

t(v)fgg

hF\(FvFw)

t(v)

f

h

h ,

where the second factor remains bounded and bounded away from zero. Using (4.2) we

therefore obtain fFv\Fw

tXf , f =

gFw\Fv

tXg, g , (5.4)

where the factor remains bounded and bounded away from zero. It follows from y0 CFwthat yf remains bounded away from zero for every f Fv \ Fw, and because xf remains

bounded, tf = yf/xf remains bounded away from zero. On the other hand it follows fromx0 Fv that, for each g Fw \ Fv, xg remains bounded away from zero, and because

yg remains bounded, it follows that tg = yg/xg remains bounded. Because Xg, < 0

for every g Fw \ Fv, it follows that the right hand side in (5.4) remains bounded, and

therefore the left hand side as well. Because Xf, > 0 for every f Fv \ Fw, and each

tf, f Fv \ Fw, remains bounded away from zero, it follows that tf remains bounded for

every f Fv \ Fw. This in turn implies that, for each f Fv \ Fw, xf = yf/tf remains

bounded away from zero, because yf does so. Therefore x0 CFw, and because also

y0 CFw, it follows that the t remain in a compact subset of NC because the NC-action

on CFw is proper.


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Therefore the NC-orbit space

Mtoric

:= CFV/NC (5.5)

has the unique structure of a Hausdorff complex analytic manifold of complex dimension

n such that the canonical projection from CFV onto Mtoric exhibits CFV as a principal NC-

bundle over Mtoric. On Mtoric we still have the complex analytic action of the complex

Lie group group TFC

/NC, which is isomorphic to the complexification TC of our real torus

T induced by the projection . The complex analytic manifold Mtoric 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 , where CF is such that

f = 1 for every f F \ Fv. Let Sv : CFv CFv be defined by Sv(z

v)f = zvf when f F

and Sv(z

v

) = 1 when f F \ Fv, as in Audin [2

, p. 159]. If Pv :CF

v CF

v /NC

denotesthe canonical projection from CFv onto the open subset Mtoricv := C

Fv /NC ofM

toric, then

v := Pv Sv is a complex analytic diffeomorphism from CFv onto Mtoricv . It is TC-

equivariant if we let TC act on CFv via TFv

Cas in Lemma 33. We use the diffeomorphism

v := v1 from Mtoricv onto C

Fv as a coordinatization of the open subset Mtoricv of

Mtoric.

If v, w V, then

Utoricv, w := v(Mtoricv M

toricw ) = C

Fv v1 w(C

Fw )

= {zv CFv | (zv)f = 0 for every f Fv \ Fw}. (5.6)

Moreover, with a similar argument as for Lemma 41, actually much simpler, we have

that for every zv Utoricv, w the element zw := w v

1(zv) CFw is given by

zwg =

fFv

(zvf)(w)

g

f , g Fw, (5.7)

where we define (zvf)0 = 1 when zvf = 0, which can happen when f Fv Fw. In this way

the coordinate transformation w v1 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 (5.7) 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 52. Mtoric

is simply connected.

Proof. Let w V. It follows from (5.6), for all v V, that the complement ofMtoricw 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 Mtoricw , after which it

can be contracted within Mtoricw to the base point in Mtoricw .

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


15/24


Theorem 53. The identity mapping from Z into CFV, followed by the canonical pro-

jectionP fromCFV to Mtoric = CFV/NC, induces a T-equivariant diffeomorphism from

M = Z/N onto Mtoric. It follows that each NC-orbit inCFV 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

CFV/NC induces a mapping : Z/N C

FV/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 ofMtoric. Because Mtoric isconnected, the conclusion is that (M) = Mtoric, that is, is surjective.

Because is a surjective submersion, dimR M = 2n = dimR Mtoric, and M is con-

nected, we conclude that is a covering map. Because Mtoric is simply connected, see

Lemma 52, we conclude that is injective, that is, is a diffeomorphism.

Remark 54 Theorem 53 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 [8, 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 f, 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 toric :=

(1)() on Mtoric will depend on the choice of RF. On the symplectic manifold

(Mtoric, toric ), the action of the maximal compact subgroup T of TC is Hamiltonian,

with momentum mapping equal to

toric := 1 : Mtoric t, (5.8)

where toric (Mtoric) = , where we note that in (2.1) depends on .

In the following lemma we compare the reduced phase space coordinatizations with

the toric variety coordinatizations.

Lemma 55. Letv V. Then Mtoricv = (Mv), and

v := v1 v (5.9)

is aTFv -equivariant diffeomorphism from Uv onto CFv .

For each zv Uv, the element v := v(z

v) is given in terms of zv by

vf = zvf

fF\Fv

rf(v(zv))

(v)ff , f Fv, (5.10)

where the functions rf : R0 are given by (3.7). We have

v(zv) = T(v(zv)) = toric (v(

v)), (5.11)


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and zv = v1(v) is given in terms of

v by

zvf = vf fF\Fv

rf()(v)f

f , f Fv, (5.12)

coordtranstoric where is the element of equal to the right hand side of (5.11).

Proof. It follows from Lemma 33 and the paragraph preceding Proposition 34 that if

zv v(Z), then zv Uv if and only ifzvf = 0 for every f F \ Fv. That is, the set Zv

in Proposition 34 is equal to Z CFv . It therefore follows from Theorem 53 that each

NC-orbit in the NC-invariant subset CFv ofC

FV intersects the N-invariant subset Zv ofZ

in an N-orbit in Zv, that is,

Mtoricv = Pv(CFv ) = (pv(Zv)) = (Mv).

If zv Uv, then Proposition 34 implies that sv(zv)f = zvf for every f Fv and

sv(zv)f = rf(v(z

v)), f F \ Fv.

If we define t TFC

by

tf = rf(v(zv))1, f F \ Fv,

tf =

fF\Fv

rf(v(zv))

(v)ff , f Fv,

then (t sv(zv))t = 1 for every t

F \ Fv and, for every f Fv, vf := (t sv)f is equal

to the right hand side of (5.10). That is, t sv(zv) = Sv(v), see the definition of Sv in

the paragraph preceding (5.6). On the other hand, it follows from (5.1) that t NC, and

therefore

v(v) = Pv(t sv(zv)) = Pv(sv(zv)) = pv(sv(zv)) = v(zv),

that is, v = v1 v(zv).

Corollary 56. Let s be the relative interior of a face of . Then T1(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 Mtoric. If s = {v} for a vertex

v, then T1(s) = {mv} for the unique fixed point mv in M for the T-action such that

T(mv) = v.

The mapping s T1(s) is a bijection from the set of all relative interiors of

faces of onto the set of all strata of the orbit type stratificiation inM for the action

of T. If s, s then s is contained in the closure of s in if and only if T1(s) is

contained in the closure of T1(s) in M.

The domain of definition Mv of v in M is equal to the union of the S such that

mv belongs to the closure ofS in M. The domain of definition Mtoricv = (Mv) of v is

equal to the union of the corresponding strata of the T-action in Mtoric, each of which is

a single TC-orbit in Mtoric. Mv and M

toricv are open cells in M and M

toric, respectively.

Proof. There exists a vertex v of such that v belongs to the closure ofs 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 CFvG denote the set of all z CFv such that zf = 0 iff G and zf = 0 iff Fv \ G.

It follows from v = T v and (3.8) that 1v (T1(s)) is equal to Uv C

FvG with


17/24


G = Fv, s. The diffeomorphism v maps this set onto the set CFvG with G = Fv, s. Because

the sets of the form CFvG with G Fv are the strata of the orbit type stratification of

the TFv -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 v1({v}) = {0} and the fact that 0 is the unique

fixed point of the TFv -action in Uv.

Ifs and v V, then Mv belongs to the closure ofT1(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 = T1(v) of v. The last statement follows from the fact that v is a

diffeomorphism from Mtoricv onto the vector space CFv , and is a diffeomorphism from

Mv onto Mtoricv .

The stratification ofMtoric by TC-orbits is one of the main tools in the survey of Danilov

[5] on the geometry of toric varieties.

Remark 57 If v, w V, then

w v1 = w

1 v = w1 (w

1 v) v = w1 (w v

1) v.

Using the formula (5.7) for w v1, this can be used in order to obtain the formulas

(4.3), (4.4) as a consequence of (5.10). In the proof, it is used that := v(zv) = w(z

w),

|zvf| = rf() if f Fv \ Fw, and fFv

(v)ff (w)gf = (w)

gf

if f F \ Fv and g Fw.

In the following corollary we describe the symplectic form toric on the toric variety

Mtoric in the toric variety coordinates.

Corollary 58. For each v V, the symplectic form(v1)(toric ) onC

Fv is equal

to (v1)(v), where v is the standard symplectic form onC

Fv given by (3.4).

Because rf(v(zv))2 is an inhomogeneous linear function of the quantities |zvf|

2, it follows

from (5.10) that the equations which determine the |zvf|2 in terms of the quantities |vf|

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 |vf|, f Fv. In this sense the

|zvf|2, f Fv, are algebraic functions of the |vf|

2, f Fv, and substituting these in (5.10)

we obtain that the diffeomorphism v1 from CFv onto Uv is an algebraic mapping. If

is a simplex, when Mtoric is the n-dimensional complex projective space, we have anexplicit formula for v

1, see Subsection 71. 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 v1. See Subsection 72.

Summarizing, we can say that in the toric variety coordinates the complex struc-

ture is the standard one and the coordinate transformations are the relatively simple

Laurent monomial transformations (5.7). However, in the toric variety coordinates the

-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


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the complex structure in the reduced phase space coordinates, which depends on , is

given by more complicated formulas.

Remark 59 It is a challenge to compare the formula in Corollary 58 for the symplectic

form in toric variety coordinates with Guillemins formula in [8, Th. 3.5 on p. 141] and [9,

(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. Cohomology classes of Kahler forms on toric varieties

For the construction of the toric variety by gluing the CFv , v V together by means

of the gluing maps (5.7), one only needs the integral Fw Fv-matrices (w v)gf := (w)

gf

Z, g Fw, f Fv as the data. The Laurent monomial coordinate transformationsUtoricv, w U

toricw, v : z

v zw are diffeomorphisms if and only if the integral matrices (w v)

are invertible, and the gluing defines an equivalence relation if and only if the matrices

satisfy the cocycle condition (u w) = (u v) (v w) for all u, v, w V. If this holds, then

the same gluing procedure allows to glue the RFv , ZFv , TFv , and (C)Fv together to an

n-dimensional vector space t, an integral lattice tZ in t, an n-dimensional torus T, and the

complexification TC ofT, respectively, where T is the unique maximal compact subgroup

of TC. Here the exponentation t e2 i t on each coordinate defines the isomorphism

t/tZ T. For each v V and f Fv the standard Z-basis vector ef ZFv is mapped

to an element Xf t, where the Xf, f Fv, form a Z-basis of tZ. The manifold

Mtoric obtained by means of the gluing process is Hausdorff = separated in the algebraic

geometric terminology, if and only if the vectors Xf t, f F, have the fan property.

The toric variety Mtoric is compact if and only if the fan is complete, which means thatt is equal to the union of all the cones C(Fv), v V.

In the above construction, F and V are just abstract finite sets, and in particular do

not yet have the interpretation of being the set of faces and vertices, respectively, of

a Delzant polytope in t. We now will construct, for each element RF satisfying

suitable conditions, a Delzant polytope = , such that F and V can be identified as

the set of faces and vertices of , respectively. For a given RF, there is a unique

solution = v t of the linear equations Xf, + f = 0 for all f Fv. On the

other hand we have the subset = of t defined by (2.1). The condition of having a

complete fan is equivalent to the existence of a choice of fs such that is a convex

polytope in t with the v, v V as its vertices. This means that Xf, v + f > 0 for

each f F \ Fv

. These s form a convex open cone in RF. Identifying v

t with

v, we obtain for each the symplectic form toric := (1) on Mtoric, where

= is the diffeomorphism from the Delzant space M = M = Z/N onto Mtoric.

Here Z = Z denotes the level set of the momentum mapping N at the level n

().

Let [toric ] H2de Rham(M

toric) denote the de Rham cohomology class of toric . For

each f F the linear form f : CF C : z zf induces a surjective homomorphism of

tori N T, hence an isomorphism N/ ker(f|N) T, and therefore N/ ker(f|N) is a

circle group. It follows from Duistermaat and Heckman [ 6, (2.10)] that, for each f F,

[toric ]

f= de Rham(cf). (6.1)


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Here cf H2(Mtoric, Z) denotes the pullback under

1 of the Chern class of the prin-

cipal N/ ker(f|N)-bundle Z/(ker f|N) over Z/N, and de Rham denotes the canonical

mapping from H2(Mtoric, Z) to H2(Mtoric, Z) R H2(Mtoric, R) H2de Rham(Mtoric),

where the first mapping is the canonical tensor product mapping, and the second and

third arrow are the universal coefficient theorem and the de Rham isomorphism, respec-

tively.

Let and R>0. The multiplication by 1/2 is an N-equivariant diffeomor-

phism from Z onto Z, and Z

toric = Z

toric . Therefore, if denotes the

diffeomorphism from Z/N onto Z /N induced by , and := 1, then

toric =

toric . Because the diffeomorphism ofM

toric is homotopic to the identity

1, its action on H2de Rham(M

toric) is trivial, and it follows that [toric ] = [toric ] for

every R>0. In combination with (6.1) this leads to the formula

[

toric

] = fFf de Rham(cf), (6.2)where we have the identities (6.3) below for the Chern classes cf, f F. The idea of

using (6.1) has also been used by Delzant [3, p. 319] and Guillemin [8, Th. 2.7]. For a

different proof, see Guillemin [9, Th. 6.3].

The mapping Z Z T : z (z, 1) induces an isomorphism from the circle

bundle Z/ ker(f|N) over Z/N onto the circle bundle Z N T over Z/N, where

t N acts on Z T by sending (z, u) to (t z, tf u). The embedding of Z T in

CFV T induces an isomorphism from the latter circle bundle onto the circle bundle

CFV NC T over C

FV/NC = M

toric, of which the Chern class is equal to the Chern class

of the associated holomorphic complex line bundle Lf = CFV NC C over M

toric, where

t NC acts on CFV C by sending (z, u) to (t z, tf u). It follows that cf = c(Lf). The

holomorphic sections of Lf correspond to the holomorphic functions s : CFV C which

are equivariant in the sense that s(t z) = tf s(z) for every t NC and z CFV. It follows

that the restriction to CFV of the linear form f defines a holomorphic section of Lf over

Mtoric, which we also denote by f. The zeroset of f is equal to

Sf := T1(f),

the smooth complex codimension one toric subvariety of Mtoric which in each CFv chart

is determined by the equation zf = 0. Because the zeros of f are simple, Sf = Div(f),

and [Lf] = (Div(f)) = (Sf), where [Lf] H1(Mtoric, O) is the equivalence class

of holomorphic line bundles containing Lf, and : H1(Mtoric, O) H2(Mtoric, Z) is

the coboundary operator in the long exact sequence induced by the short exact sequence

0 Z Oe2 i 1. Here O and O respectively denote the sheafs of germs of holomorphic

and nowhere vanishing holomorphic functions on Mtoric, and e2 i is the homomorphism

f e2 i f from the sheaf O of additive groups to the sheaf O of multiplicative groups.

It follows that

cf = c([Lf]) = c((Sf)) = i([Sf]), (6.3)

where i : H2n2(Mtoric, Z)

H2(Mtoric, Z) denotes the Poincare duality isomorphism

defined by the intersection numbers of chains. See Griffiths and Harris [7, bottom of p.

55 and Ch. 1, Sec. 1] for for the background theory.


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Let denote the collection of F such that Fv for some v V. Then the

face := f

f, ,

are the closed faces of of arbitrary dimensions. For each we have the smooth

toric subvariety

S = T1(face)

ofMtoric, which in each CFv chart is determined by the equations zf = 0, f . It follows

from Danilov [5, Cor. 7.4, Prop. 10.3 and 10.4, and Th. 10.8] that H i(Mtoric, O) = 0

for every i > 0, H(Mtoric, Z) is generated by the homology classes of the S, ,

and H(Mtoric, Z) has no torsion. Because H2(Mtoric, O) = 0, every c H2(Mtoric, Z)

is equal to the Chern class of a holomorphic line bundle L over Mtoric, which moreover

is unique up to isomorphisms because H1(Mtoric, O) = 0. Because H2n2(Mtoric, Z)

H2

(Mtoric, Z) is generated by the [Sf], f F, we have

c(L) = c =fF

nf i([Sf]) =fF

nf c(Lf) = c(fF

Lnf ), hence L fF

Lfnf

for suitable integers nf, f F.

Let 1 denote the set of all such that dimR (S) = 2, that is, dimR (face) = 1,

or equivalently S is a complex projective line = Riemann sphere embedded in Mtoric.

Because H2(Mtoric, Z) H2n2(Mtoric, Z) has no torsion, the canonical homomorphism

de Rham : H2(Mtoric, Z) H2de Rham(M

toric) is injective, and H2de Rham(Mtoric)

belongs to the image if and only ifS

Z for every 1.

It follows in view of (6.2) that

[

toric

] de Rham(H

2

(M

toric

,Z

)) fF f (Sf S) Z 1, (6.4)where Sf S Z denotes the intersection number of the toric subvarieties Sf and S of

Mtoric. Furthermore, if [toric ] de Rham(H2(Mtoric, Z)), then [toric ] = de Rham(c(L))

for a holomorphic complex line bundle L over Mtoric, which is uniquely determined up

to isomorphisms and isomorphic to a product of integral powers of the Lf, f F.

For example, if is a simplex, when Mtoric is isomorphic to the n-dimensional com-

plex projective space CPn, then the Sf are complex projective hyperplanes. All complex

projective hyperplanes in CPn define one and the same holomorphic line bundle, called

the hyperplane bundle, the Chern class c of which generates H2(CPn, Z) Z. In this

case [toric ] = (

fF f) de Rham(c).

The nondegeneracy of toric implies that the Hermitian form h on Mtoric, of which

toric is the imaginary part, is nondegenerate at every point. At the origin in the CFv -coordinates, toric is equal to a positive multiple of (3.4) and h is positive definite there.

Because Mtoric is connected, the signature of h is constant, hence h is positive definite

everywhere. That is, toric is a Kahler form on Mtoric.

Let be an arbitrary Kahler form, a closed two-form on Mtoric equal to the imaginary

part of a positive definite Hermitian structure h on Mtoric. Because T is connected, the

pullback t() of by t T is homotopic and therefore cohomologous to . The average

h of the positive definite Hermitian forms t(h) over all t T is positive definite, and the

imaginary part of h is equal to the average of Im(t(h)) = t(Im(h)) = t() over all

t T. It follows that is a T-invariant Kahler form with the same cohomology class as .


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Because Mtoric is simply connected, see Lemma 52, we have H1de Rham(Mtoric) = 0, hence

the T-action is Hamiltonian with respect to , with a momentum map : Mtoric t.

It follows from Delzant [3, Lemme 2.2 and (*) on p. 323] that each fiber of is a single

T-orbit, = (Mtoric) is a Delzant polytope, and the pre-images under of the relative

interiors of the faces of are the connected components of the orbit types for the T-

action in Mtoric. In particular the codimension one faces and the vertices of correspond

bijectively to the f F and v V used in the gluing construction ofMtoric. For any f

F, let Yf denote the infinitesimal action ofXf t on Mtoric. Because is a Kahler form,

we have in the complement of the zeroset of Yf that 0 < (Yf, J Yf) = dXf, (JYf),

where J denotes the complex structure. If v V, f Fv, and we write the CFv -

coordinates as zf = xf + i yf with xf, yf R, then 2 Yf = yf /xf + xf /yf, and

therefore minus its J -image is equal to xf /xf+ yf /yf. It follows that in the radial

direction xf /xf + yf /yf the function Xf, is increasing close to zf = 0, hence

is contained in the half spaceX

f,

+

f 0 if

X

f,

+

f= 0 on the face of

corresponding to f. Therefore there exists a such that is equal to the polytope

defined by (2.1).

Because (Mtoric) = = = (Mtoric), it follows from Delzant [3, Th. 2.1] that

there exists a T-equivariant diffeomorphism of Mtoric such that = (toric ) and

= . As the fibers of the momentum mappings are the T-orbits, is both T-

equivariant and preserves the T-orbits. Because 1(int) is the set on which the action

of T is free, it follows that for every int there is a unique t = () T, such

that (m) = t m for every m 1({}). A straightforward analysis of the equation

(m) = ((m)) m shows that the function : int T extends to a smooth T-valued

function on , which we also denote by . Because is simply connected, has a lift

to the covering t of T, that is, there exists a smooth mapping : t such that

() = exp(()) for every . The mappings s : m exp(s ((m))) m form asmooth homotopy of diffeomorphisms, where 1 = and 0 = 1, the identity in M

toric.

Therefore the action of on the de Rham cohomology groups is trivial, and we conclude

that [] = [] = ([toric ]) = [toric ]. In view of (6.2) and (6.3) this leads to the identity

{ [] | is a Kahler form on Mtoric } = {fF

f i([Sf]) | } (6.5)

between subsets of H2de Rham(Mtoric). The set on the left and right hand side of (6.5) is

equal to the open convex cone in Delzant [3, Prop. 3.2].

As averaging of a symplectic form does not necessarily yield a symplectic form, we

do not make a statement about the cohomology classes of arbitrary symplectic forms

on Mtoric. We also note that if is a T-invariant symplectic form on Mtoric, then it is

a Kahler form on Mtoric with respect to some complex structure on Mtoric, which ingeneral is very different from the one we started out with. For instance, negative toricis not a Kahler form with respect to the initial complex structure on Mtoric.

7. Examples

71. The complex projective space

Let the Delzant polytope be an n-dimensional simplex in t. That is, there are

numberings fi and vi, 0 i n, of F and V, respectively, such that Fvj = {fi | i = j}

for each j. Write ei = Xfi . Then the ei, 1 i n, form a Z-basis of the integral lattice


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tZ in t, and there are mi Z such that e0 =n

i=1 mi ei. For each 1 j n the vectors

ei such that i = j form a Z-basis of tZ, which condition is equivalent to mj = 1. The

fan property implies that the intersection of the cone spanned by this basis with the cone

spanned by the e1, . . . , en is contained in a hyperplane, hence mj 0. The conclusion

is that

e0 = n

i=1

ei. (7.1)

In the sequel we identify fi with its number i, whence CF = {(z0, . . . , zn) | zi C}. The

Delzant simplex (2.1), determined by the inequalities ei, + i 0, 0 i n, has a

non-empty interior if and only if

:=

ni=0

i > 0. (7.2)

In the sequel we take v = v0, that is, ei, v + i = 0 for all 1 i n. If we write

i = ei, , 1 i n, when t, then (3.2) yields that

v(zv)i = |zi|

2/2 i, 1 i n.

It follows from (3.7) that

r0() = (2(n

i=1

i + 0))1/2,

and therefore (5.10) yields that

vi = zvi (2 z

v2)1/2, 1 i n, (7.3)

where we have written

zv2 =n

i=1

|zvi |2.

Note that Uv is the open ball in Cn with center at the origin and radius equal to (2c)1/2.

The equations (7.3) imply that

v2 = zv2/(2 zv2),

hence

zv2 = 2v2/(1 + v2).

Therefore the mapping v1 : v zv is given by the explicit formulas

zvi = vi (2/(1 +

v2))1/2, 1 i n. (7.4)

We have CFV = Cn+1 \ {0}, nC is the set of all t Cn

+1 such that ti = t0 for every

1 i n, and we recognize the toric variety as the quotient ofCn+1 \{0} by the action of

multiplications by nonzero complex scalars. That is, the toric variety is the n-dimensional

complex projective space CPn. The symplectic form toric = (v1)(v), where v is the

standard symplectic form (3.4), is equal to times the Fubini-Study form as defined in

Griffiths and Harris [7, p. 30, 31]. As we have see in Section 6, the de Rham cohomology

class of toric is equal to times the de Rham cohomology image of the Chern class of

the hyperplane bundle. This corresponds to the classically known fact that integral of the


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Fubini-Study form over any complex projective line in CPn is equal to one, see Griffiths

and Harris [7, p. 122].

72. The Hirzebruch surface

Let n = 2. Then k := #(F) 3 and the faces form a cycle fi, i Z/kZ, with vertices

vi fi1 fi, when Fvi = {fi1, fi}. Write ei = Xfi . Then ei+1 = bi ei1 + di ei for

bi, di Z. Because the matrix

Mi =

0 bi1 di

has an integral inverse, we have bi = 1. The fan property implies that the cone

spanned by ei1 and ei intersects the cone spanned by ei and ei+1 in R0 ei, hence

bi 0, and therefore bi = 1. The cocycle condition is equivalent to the condition that

Mk Mk1 . . . M 2 M1 = 1.

For k = 3 the cocycle condition leads to d1 = d2 = d3 = 1, and we recover thecomplex projective plane as in Subsection 71. For k = 4, the case discussed in this

subsection, the cocycle condition is equivalent to the equations 1 d2 d3 = 1, d1 + (1

d1 d2) d3 = 0, d2 + (1 d2 d3) d4 = 0, and 1 d1 d2 (d1 + (1 d1 d2) d3) d4 = 1. The

solutions to these equations are d1 = d3 = 0, d4 = d2, or d2 = d4 = 0, d3 = d1,

when is a parallelogram or a trapezium. By means of a cyclic shift and/or an inversion

in the numbering of the faces we can arrange that e3 = e1 + m e2 and e4 = e2 for

some m Z0, and we recognize the toric variety, obtained by gluing four copies of

C2 together by means of the Laurent monomial coordinate transformations (5.7), as the

Hirzebruch surface m, see Hirzebruch [10].

The Delzant polytope (2.1) is determined by the inequalities ei, +i 0, 1 i 4,

which is a quadrangle if and only if

:= 1 + 3 m 4 > 0, (7.5)

and the inequalities imply that 2 + 4 = + + > 0.

In the sequel we take for v the vertex determined by the equations ei, + i = 0 for

i = 1, 2, where Fv = {1, 2}. If we write i = ei, , 1 i 2, when t, then (3.2)

yields that

v(zv)i = |zi|

2/2 i, 1 i 2.

It follows from (3.7) that

r3() = (2(1 + m 2 + 3))1/2, r4() = (2(2 + 4))

1/2

and therefore (5.10) yields that

v1 = zv1 (2|z

v1 |

2 + m |zv2 |2)1/2, (7.6)

v2 = zv2 (2|z

v1 |

2 + m |zv2 |2)m/2 (2(+ + ) |z

v2 |

2)1/2. (7.7)

If we write ti = |zvi |2 and i = |vi |

2, then this leads to the equations

1 = t1/(2 t1 + m t2),

2 = t2 (2 + t1 + m t2)m/(2(+ + ) t2)

for t1, t2. If we solve t1 from the first equation,

t1 = (2 + m t2) 1/(1 + 1),


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

(1 + 1)m

2 (2(+ + ) t2) = t2 (2 + m t2)m

(7.8)

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 1, 2 R0 there is a unique solution t2 R0, confirming the first statement in

Lemma 55.

On the other hand, if we work overC, and view both the parameter := (1+1)m 2 and

the unknown t2 as elements of the complex projective line CP1, then the equation (7.8)

defines a complex algebraic curve C in the (t2, )-plane CP1 CP1, where the restriction

to C of the projection to the first variable t2 is a complex analytic diffeomorphism from

C onto CP1, as on C we have that is a complex analytic function of t2. In particular

C is irreducible. The restriction to C of the projection to the second variable is an

(m + 1)-fold branched covering. Over = 0 and over =

we have that m of the m + 1

branches come together, whereas there are two more branch points on the -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 (7.8), as a complex analytic function of in

the complement of the branch points, will reach each and every other branch if runs

over a suitable loop. In other words, the solution t2 is an algebraic function of of degree

m + 1, and no branch of a solution is of lower degree. This holds in particular for our

solutions t2 R0 for R0.

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. Birkhauser, Basel,Boston, Berlin, 1991.

[3] T. Delzant: Hamiltoniens periodiques et images convexes de lapplication moment. Bull.Soc. Math. France 116 (1988) 315339.

[4] M. Demazure: Sous-groupes algebriques de rang maximum du groupe de Cremona. Ann.

scient. Ec. Norm. Sup. 3 (1970) 507588.[5] V.I. Danilov: The geometry of toric varieties. Russ. Math. Surveys 33:2 (1978) 97154,

translated from Uspekhi Mat. Nauk SSSR 33:2 (1978) 85134.[6] J.J. Duistermaat and G.J. Heckman: On the variation in the cohomology of the symplectic

form of the reduced phase space. Invent. math. 69 (1982) 259268.[7] P. Griffiths and J. Harris: Principles of Algebraic Geometry. J. Wiley & Sons, Inc., New

York, etc., 1978.[8] V. Guillemin: Moment Maps and Combinatorial Invariants of Hamiltonian Tn-spaces.

Birkhauser, Boston, etc., 1994.[9] V. Guillemin: Kaehler structures on toric varieties. J. Differential Geometry 40 (1994)

285309.[10] F. Hirzebruch: Uber eine Klasse von einfach-zusammenhngenden komplexen Mannig-

faltigkeiten. Mathematische Annalen124 (1951) 7786.[11] A. Pelayo: Topology of spaces of equivariant symplectic embeddings. Proc. Amer. Math.

Soc. 135 (2007) 277288.[12] R.T. Rockafellar: Convex Analysis. Princeton University Press, princeton, N.J., 1970.

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