Transformation Groups: Symplectic Torus Actions and Toric Manifolds

Transformation Groups

Symplectic Torus Actions and Toric Manifolds

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Transformation Groups

Symplectic Torus Actions and Toric Manifolds

Edited by Goutam Mukherjee

With Contributions by

fl::[gl@ODHINDUSTAN U LUJ UBOOKAGENCY

Chris Allday Mikiya Masuda

P Sankaran

Editor:

Goutam Mukherjee Indian Statitical Institute Kolkata India [email protected]

Contributors:

Chris AI:day University of Hawaii USA [email protected]

Published By

Hindustan Book Agency (India) P 19 Green Park Extension New Delhi 110016 India

email: [email protected] http://www.hindbook.com

Mikiya Masuda Osaka city university Japan [email protected]

Parameswaran Sankaran Institute of Mathematical Sciences Chennai India [email protected]

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ISBN 978-81-85931-54-8 ISBN 978-93-86279-30-9 (eBook)DOI 10.1007/978-93-86279-30-9

Contents

Prefaee vii

1 Loealization Theorem and Sympleetie Torus Aetions 1 1.1 Introduction......... 1 1.2 The Borel Construetion . . 4 1.3 The Loealization Theorem. 1.4 Poineare Duality . . . . . . 1.5 A Brief Summary of Symplectie Torus Aetions 1.6 Cohomology Sympleetie and Hamiltonian Torus

Aetions ... 1.7 An Example .

2 Torie Varieties 2.1 Introduetion ...... . 2.2 Affine torie varieties .. 2.3 Fans and Torie Varieties 2.4 Polytopes . . . . . . . . 2.5 Smoothness and Orbit Strueture 2.6 Resolution of singularities. . . . 2.7 Complete nonsingular torie surfaees 2.8 Fundamental Group . . 2.9 The Euler eharaeteristic . . . 2.10 Line bundles ........ . 2.11 Cohomology of torie varieties 2.12 The Riemann-Roeh Theorem 2.13 The moment map .....

3 Torus aetions on manifolds 3.1 Introduetion ...... . 3.2 Equivariant eohomology

9 20 28

35 41

43 43 44 47 52 53 61 63 66 68 70 76 79 82

85 85 88

VI

3.3 Representations of a torus ......... . 3.4 Torie manifolds . . . . . . . . . . . . . . . . 3.5 Equivariant eohomology of torie manifolds . 3.6 Unitary torie manifolds and multi-fans 3.7 Moment maps and equivariant index 3.8 Applieations to eombinatories

Bibliography

Index

CONTENTS

· 93 · 96 · 101 · 107 · 114 · 117

122

129

Preface

This volume is updated and revised version of the main lectures delivered in the Winter School on Transformation Groups held in honor of Professor Amiya Mukherjee at the Indian Statistical Institute, Kolkata in December 1998 under the auspices and financial support of the Indian Statistical Institute, the National Board of Higher Mathematics, Chennai Mathematical Institute and the Council for Industrial and Scientific Research, India.

The aim of the school was to discuss the recent trends in the use of cohomological methods in the study of torus actions and toric manifolds and to expose young topologists to the recent developments in these areas.

A very interesting aspect of the cohomology theory of transformation groups is its interaction with the study of symplectic and Hamiltonian torus actions. Many of the results of the latter theory are cohomological. The importance of cohomology theory in the study of symplectic and Hamiltonian torus actions has been recognized for a long time and the usefulness of cohomology theory in the field continues today, significantly in the theory of toric varieties.

Chapter 1 is devoted to illustrate the cohomological methods used in the study of symplectic and Hamiltonian torus actions.

The basic theory of toric varieties was established in the early 70's by Demazure, Mumford etc., and Miyake-Oda. It says that there is a oneto-one correspondence between toric varieties and combinatorial objects called fans. Moreover, a compact non-singular toric variety together with an ample complex line bundle corresponds to a convex polytope through a map called the moment map. Chapter 2 is abrief introduction of the basic theory of toric varieties.

Finally, chapter 3 is intended to develop the theory of toric varieties, which is a bridge between algebraic geometry and combinatorics, from a topological point of view. This is done by studying new geometrical objects called toric manifolds, which generalize many results of toric varieties in a topological framework and produce nice applications relating

VIU Preface

topology, geometry and combinatorics. Most of the techniques and proofs of results given in the notes are

either new and have not appeared elsewhere in the literat ure or are written in a style which may be more accessible to the readers.

C. Allday, M. Masuda, G. Mukherjee, P. Sankaran.

About the Notes

This volume is updated and revised version of the main lectures delivered in the Winter School on Transformation Groups held at the Indian Statistical Institute, Kolkata in December 1998. The Chapter 1 of this volume is written by Professor Christopher Allday, Chapter 2 is written by Professor Parmeswaran Sankaran and Chapter 3 is written by Professor Mikiya Masuda. Dr. Goutam Mukherjee organizesd the school and acted as a coordinating editor.

Acknowledgements

This volume is based on lectures delivered by Professor Christopher Allday, University of Hawaii, U.S.A, Professor Mikiya Masuda, Osaka City university, Japan and Professor Parameswaran Sankaran, Institute of Mathematical Sciences, Chennai, India, at the winter school on 'Transformation Groups' that was held at Indian Statistical Institute, Kolkata, India, from 8th december to 26th December, 1998. This was one of aseries of meetings of topologists in India that are being organized at various Institutions in India since 1987 starting at NEHU, Shillong. I express my sincere thanks to Professor B. 1. Sharma, for initiating this idea and being with us throughout.

I extend my warm and sincere thanks to Professor Chris Allday, Professor Mikiya Masuda and Professor Parameswaran Sankaran on my own behalf as weIl as on behalf of everyone else involved in the project, for accepting our invitation to visit Kolkata, for delivering such superb lectures at the school and writing not es for this volume. I express my sincere thanks to all other speakers who delivered Iectures to make the audience prepared for the main Iectures.

A course at this level cannot succeed without contributions from everyone involved in the project and I thank aIl my colleages and staff of ISI, Kolkata for their help in organizing the schooI. Specially I thank my friend and colleague Dr. A. C. Naolekar without his enormous help and cooperation it would not have been possible for me to achieve this

Preface IX

goal. I wish to re cord my sincere thanks to Professor M. S. Raghunathan and Professor C. S. Seshadri for extending financial support. I also thank CSIR for providing financial support. I express my sincere thanks to Professor S. B. Rao, the then Director of ISI and Professor S. C. Bagchi, the then Professor-in-charge, Stat-Math division, ISI for their support. I also thank Hindustan Book Agency, for accepting this work for publication.

Goutam Mukherjee We warmly thank Prof. Goutam Mukherjee for inviting us to speak

in the winter school on Transformation Groups held at ISI, Kolkata, in December 1998 and for his interest in these notes. But for his sustained and patient efforts, these notes would not have been published. And we thank ISI, Kolkata, for the warm and generous hospitality extended to us during the winter school.

Chris Allday, Honolulu, Mikiya Masuda, Osaka,

Parameswaran Sankaran, Chennai.

Chapter 1

Localization Theorem and Symplectic Torus Actions

1.1 Introduction

These notes are intended to be abrief introduction to the cohomology theory of transformation groups with applications to symplectic and Hamiltonian torus actions. More ar less without exception - see, e.g., Theorem 1.6.5 - we consider only torus actions, and, often only circle actions. The main theorem of the subject is the Localization Theorem (Theorem 1.3.7), and, except in the more general statement and proof of this theorem, we use only cohomology with coefficients in a field of characteristic zero. Also, to furt her simplify the presentation, we prove the Localization Theorem only in the compact case, discussing other cases in remarks. As in the original Smith theory, the cohomology theory is useful too for studying elementary abelian p-group actions; but that will not be done here: see [4] far a more comprehensive introduction. For an introduction to compact Lie group actions in general, see [14], [28] or [56].

A very interesting aspect of the cohomology theory of transformation groups is its interaction with the study of symplectic and Hamiltonian torus actions. Many of the results of the latter theory are cohomological; and it is interesting to see which of them follow from the Localization Theorem and which of them do not. In these notes we give several results of each type. Along the way, in Section 1.5, we give the basic definitions of the theory of symplectic and Hamiltonian group actions. The definitions, while quite complicated for compact connected Lie group actions in general, become simple and elegant for circle actions.

The importance of cohomology theory in the study of symplectic and

2 1. Localization Theorem and Symplectic Torus Actions

Hamiltonian torus actions has been recognized for a long time - see, for example, [11 J and [30J - and the usefulness of cohomology theory in the field continues today, significantly in the theory of toric varieties.

In Section 1.2 we discuss the Borel construction and some properties of the Leray-Serre spectral sequence which are used subsequently. Section 1.3 states and proves the Localization Theorem and gives so me immediate corollaries. Section 1.4 concerns the consequences of Poincare duality. Some of the basic results here are stated without proof, since the proofs are rather long; and they can be found in [4J, for example. But we do give proofs of the Topological Splitting Principle and the Borel Formula: these show so me of the power of the Localization Theorem. Seetion 1.5 introduces symplectic and Hamiltonian group actions with emphasis on Frankel's results. Arecent theorem of Jones and Rawnsley is included also. In Section 1.6 we give the cohomology analogues of the definitions in the symplectic and Hamiltonian theory, and, again, we illustrate the power of the Localization Theorem with results such as Theorems 1.6.5 and 1.6.10. The brief and final Section 1.7 shows by example that a theorem of McDuff and recent theorems of Tolman and Weitsman and Giacobbe are not purely cohomological.

For the remainder of this introduction we give some of the basic terminology of transformation groups, some comments on the cohomology theory and the spectral sequence used in these notes, and so me other assumptions and conventions.

Definition 1.1.1. Let X be a HausdorJJ topological space and let G be a compact Lie group.

(1) An action of G on X on the left is a map .p : G x X ~ X such that .p(I, x) = x, for all x E X, where 1 E G is the identity, and .p(g, .p(h, x)) = .p(gh, x), for alt x E X and all g, h E G. Usually .p(g, x) is written simply as gx; and so the two conditions become Ix = x and g(hx) = (gh)x for all x E X and all g, h E G.

An action on the right is defined similarly. In these notes, unless indicated otherwise, actions will be on the left.

(2) Given any action of G on X and given x E X, the isotropy subgroup of x is Gx = {g E G; gx = x}; and the orbit of x is G(x) = {gx; gE G}

For any y E G(x), clearly, Gy is conjugate to Gx . And there is a homeomorphism G/Gx ~ G(x) given by gGx t-----+ gx.

1.1. Introduction 3

(3) The fixed point set 01 an action 01 G on X is

X G = {x E X;gx = x,/or alt gE G}

= {XEX;Gx=G}.

(4) An action 01 G on X is said to be Iree il Gx = 1, the trivial subgroup 01 G, lor alt x E X: i.e., illor every x E X, gx = x implies that 9 = 1. The action is said to be ~ il Gx is finite lor all x EX. And the action is said to be ~ il X G =I- 0 and the action is free on X - X G : i. e., if X G =I- 0 and G x = 1 lor all x 1. X G .

(5) An action 01 G on X is said to be effective il gx = x lor all x E X implies that 9 = 1: i.e., if n Gx = 1. In general, n Gx is called

xEX xEX the ineffective kernel. Clearly it is anormal subgroup 01 G.

The action is said to be ~ il the ineffective kernel is finite.

(6) The relation x"" y il and only ify E G(x) is an equivalence relation on X. The equivalence class [xJ = G(x). The quotient space is denoted X / G, and it is called the orbit space. The quotient map 1f : X ---+ X / G is calted the orbit map.

The orbit space is HausdorjJ, and the orbit map is open, closed and proper: see, e.g., !14}, Chapter I, Theorem 3.1 .

(7) The space X is called a ~ when there is given an action 01 G onX.

For the existence and properties of tubes and slices see, for example, [14J, Chapter II, Sections 4 and 5, and, for the smooth case, see [14], Chapter VI, Section 2, especially Corollary 2.4 .

To prove the Localization Theorem one wants a cohomology theory with the tautness property for closed sets. Since we shall consider only paracompact spaces, Alexander-Spanier cohomology is a good choice. The tautness property used is the following. Let X be a paracompact space and let A ~ X be a closed subspace. Let N be the system of open neighborhoods of A, or the system of closed neighborhoods of A, directed downward by inclusion. Let H*( -; k) denote Alexander-Spanier


cohomology with coefficients in an abelian group k. Then restriction induces an isomorphism

lim H*(N; k) ~ H*(A; k). ---+ NEN

See [76], Chapter 6, Section 6 or [15J, Chapter II, Section 10 for the proof.

Thus, throughout these notes, the cohomology theory used will be Alexander-Spanier cohomology. Alexander-Spanier cohomology coincides with Cech cohomology for all spaces; and, for paracompact spaces, it coincides with sheaf cohomology and the cohomology theory given by the Eilenberg-MacLane spectrum. For paracompact locally contractible spaces, Alexander-Spanier cohomology coincides with singular cohomology. So singular theory works weIl for smooth actions on smooth manifolds or, more generally, on G - CW -complexes.

Because Alexander-Spanier cohomology, or, equivalently, sheaf cohomology , since all spaces will be paracompact, is used, in order to study the cohomology of fibre spaces, one needs the Leray spectral sequence (see [15]) or, as it is called in this context, the Leray-Serre spectral sequence. This is the same as the Serre spectral sequence when singular cohomology is used. [76], Chapter 9, is a good introduction. See also [66J.

All spaces in these notes will be assumed to be Hausdorff. And the terms 'compact' and 'paracompact' will be assumed to include Hausdorffness as part of their definitions.

Aigebraic notations and conventions(l) Elements in graded rings will be assumed to be homogeneous unless mentioned otherwise. Similarly, ideals in graded rings will usually be homogeneous. The prime ideals, 13, which occur from time to time in Section 1.3, however, are not assumed to be homogeneous.

(2) If R is a commutative ring, M is aR-module, and 13 ~ R is a prime ideal, then Rp, respectively Mp, denotes R, respectively M, localized at 13, i.e., with respect to the multiplicative set R - p. In particular, if R is an integral domain, then R(o) is the field of fractions.

1.2 The Borel Construction

Let G be a compact Lie group. There is a universal principal G-space EG: that is, EG is a contractible paracompact space on which G acts

1.2. The Borel Construction 5

freely on the right. The Milnor method is a way of constructing EG functorially: see [52], Chapter 4, Seetion 11. It is often more convenient, however, to think of EG differently. Since G has faithful representations, G can be embedded homomorphically in so me U (m) and some O(n). Thus EU(m) or EO(n) can serve as EG. And these are quite explicit: EU(m) = Vm(Coo) and EO(n) = Vn(ROO ) , the Stiefel spaces of m-frames, respectively n-frames, in infinite complex, respectively real, space. Futhermore, these Stiefel spaces are direct limits of the closed Stiefel manifolds in an obvious way. (See, e.g., [66], Section 6.2.1, for more, and for some interesting references.) It is useful that Vm(Coo ) and Vn(Roo ) are CW - complexes.

When G = SI, the circle group, one can take EG = SOO, the direct limit, via inclusion, of the unit spheres in cn as n tends to infinity. And, for G = T r = Si x··· X Si (r copies), one can take EG = SOO x··· x SOO (r copies).

The classifying space, BG, is the orbit space EG jG. And there is the universal principal bundle G ~ EG ~ BG.

Example 1.2.1. One has the following cohomology rings. ([12].)

(1) For G = Si, H*(BG; Z) = Z[t], where deg (t) = 2.

(2) For G = T r , H*(BG; Z) = Z[tl,"" tr ], where deg (ti) = 2 for l::;i::;r.

(3) For G = U(n), H*(BG; Z) = Z[cl,"" cn ], where CI, ... ,Cn are the universal Chern classes of degrees 2, ... ,2n, respectively.

Furthermore, one has the following useful fact.

(4) If G is a compact connected semi-simple Lie group, then Hi(BG; Q) = 0 for 1 ::; i ::; 3.

To see this, one looks at the homotopy long exact sequence for G ~ EG ~ BG. Since EG is contractible, 7ri(BG) ~ 7ri-l(G). So BG is simply connected, because G is connected. Also 7r3(BG) is trivial, because 7r2(G) is trivial for any compact connected Lie group. (See, e.g., [52], Chapter 7, 12.4.) Finally, 7r2(BG) is finite, because 7rl(G) is finite for semi- simple G. (See, e.g., [14], Chapter 0.)

Definition 1.2.2. Let G be a compact Lie group, and let X be aspace with G acting on the left. Let G act diagonally on EG x X,' i. e., g( z, x) = (zg-l,gx) for all gE G, z E EG and xE X. Let Xa = (EG x X)jG,


the orbit space. This is the Borel construction. (The orbit space Xc is olten also denoted by EG -+ xX.) Furthermore, given an abelian

c group or commutative ring 01 coefficients, k, one defines the equivariant cohomology 01 X with coefficients in k, Hc(X; k), to be H*(Xc; k).

Remark 1.2.3.

(1) If G acts freely on a paracompact space X, then the map q: Xc ---? X/G, [z, x] f------t [x], is a homotopy equivalence.

If all the isotropy subgroups are finite, i.e. G acts almost-freely on X, and if k is a field of characteristic 0, then q* : H*(X/G; k) ---?

Hc(X; k) is an isomorphism. (See, e.g., [72].)

(2) Since one might want to use different choices of EG in different contexts, Xc is not canonically defined. But any two choices of EG yield homotopy equivalent Borel constructions if X is paracompact. (See [72], Section 1, and [28], Chapter 1, Proposition (8.18).)

(3) If G is a compact Lie group and K ~ G is a closed subgroup, then G acts transitivelyon the homogeneous space G / K; and the map

(G/K)C = EG Xc G/K ---? EG/K = BK,

given by [z, gK]c = [zg, K]c f------t [zg]K, is a homeomorphism, as it is easy to see.

In particular, if G is acting on aspace X, then, for any x EX, G(x)c ~ BGx ·

The Borel construction is, by definition, the total space of the associated fibre bundle p : Xc ---? BG, given by [z, x] f------t [z], with fibre X. If G is connected, then BG is simply connected (see Examples 1.2.1 (4) above); and so the Leray-Serre spectral sequence for p has constant coefficients in the Ez-term. In particular, taking cohomology with coefficients in a field, k, the Leray-Serre spectral sequence has the form

The edge effects of this spectral sequence determine

p* : H*(BG; k) ---? Hc(X; k)

1.2. The Borel Construction 7

and i* : Hc(X; k) ----+ H*(X; k), where i : X ----+ Xc is inclusion of the fibre. The spectral sequence collapses, i.e., E;'* = ~ if and only if i* is surjective. In this case, i.e., collapse, one says that X is totally non-homologous to zero (TNHZ) in Xc ----+ BG. (This terminology makes sense, since the map i* in homology must be injective.)

Let R = H*(BG; k). Then the map p* makes Hc(X; k) into a Rmodule and aR-algebra. (When G is connected and k is a field of characteristic 0, R is zero in odd degrees; and so R is commutative in the usual sense.) For a E Rand x E Hc(X; k), we shall denote this module (algebra) multiplication by ax: i.e., ax = p*(a)x, where the latter product is the cup product in Hc(X; k). (One must be careful with this notation if p* is not injective: for then, for example, for a E R and 1 E ~ k), one has al = p*(a), not a. If p* is injective, then it is safe to identify p* (a) with a.)

The usual filtration on Hc(X; k) in the Leray-Serre spectral sequence for p : Xc ----+ BG is the basewise or decreasing filtration: thus

EP,q ~ 'L Hp+q(X' k)/F IHp+q(X' k) 00 - Jp C ' p+ C ,.

But, in studying equivariant cohomology , it is usually much more helpful to consider the increasing or fibrewise filtration defined by 'LqHp+q(X' k) = F Hp+q(X' k) Thus .r c , pc ,.

So

The reason why the increasing filtration is useful is that each row E;,q in the spectral sequence is aR-module (where, as above, R = H*(BG; k)); each differential, dr is aR-module homomorphism (since dr is zero when restricted to E;'o, which is a quotient of E;'o = R); and FqHc(X; k) is a R-submodule of Hc(X; k).

There is a second bundle associated with the Borel construction, namely the principal bundle EG x X ----+ Xc with fibre G. Of course, there is a homotopy equivalence EG x X ::::= X. Thus, up to homotopy, one can think of this bundle as i : X ----+ Xc, the inclusion of the fibre in the bundle p. And it is easy to see that this principal G-bundle is classified by p. That is, there is a pull-back


EGxX ~ EG

1 1 ~

Xc p BG.

where the top horizontal map is the projection, and the right vertical map is the universal principal G-bundle, EG ~ EG jG = BG.

Thus, when G is connected, one also has constant coefficients in the ETterm of the Leray-Serre spectral sequence of G ~ X ~ Xc: see, e.g. , [76], Chapter 9, Section 2, Theorem 5.

Another context giving rise to a pull- back is the following. Let G be a compact Lie group, let X be a G- space, let K ~ G be a closed subgroup, and let K act on X via the inclusion into G (Le., the G- action restricted to K). We can let EG serve as EK also; and so there is a map XK ~ Xc given by [z, XJK I----t [z, xJc, and a map BK ~ BG given by [ZJK I----t [zJc. With these maps, the following diagram is a pull- back (as is easily seen):

1 1 BK ~ BG.

By the naturality of the Leray- Serre spectral sequence, this diagram gives rise to a useful map of spectral sequences.

We conclude this section with three comments.

Remark 1.2.4.

(1) Consider the Leray-Serre spectral sequence for p : Xc ~ BG when, as above, G is connected and we take coefficients in a field, k, of characteristic O. As remarked above Hi(BG; k) = 0 if i is odd. Now suppose that Hi(X, k) = 0 when i is odd also. Thus ~ = 0 if either p or q is odd. But dr sends Ef,q to Ef+r,q-r+l :

thus dr = 0 for all r ~ 2; and hence E;'* = ~ In other words (see above), X is TNHZ in Xc ~ BG if Hi(X; k) = 0 for all odd i. (In the latter case, i.e., Hi(X; k) = 0 for all odd i, one says that H*(X; k) is evenly graded.)

(2) Even when G is connected, some of the isotropy subgroups, Gx ,

may fail to be connected. So we need to briefly consider nonconnected groups. Let G be a compact Lie group, and let CO be

1.3. The Localization Theorem 9

the component of the identity. So GO is a closed normal subgroup of G; and G IGo is finite. It follows from the Leray- Serre spectral sequence for the bundle BG ----+ B( G IGO) with fibre BGo, or by considering the considering the covering map c : BGo ----+ BG with fibre GIGo , that cinduces an isomorphism

where the latter is the fixed algebra of an action of GI GO on H*(BGo; k) which is induced by G acting on GO by conjugation, and k is a field of characteristic o. In particular, c* : H* (BG; k) ----+ H*(BGo; k) is injective.

If G is abelian, then GO is a torus; and GI GO acts triviallyon H*(BGo; k) . Thus c* : H*(BG; k) ----+ H*(BGo ; k) is an isomorphism when G is abelian. (If G is a subgroup of a torus, T, then this also follows because thell T ~ GO x TI GO.)

(3) Let G be a cOlllpact Lie group, alld let N be a positive integer. Then, by considering the compact Stiefel manifolds inside an appropriate (Stiefel) choice for EG, there is a compact free Gsubspace Etf ~ EG such that Hi(Etf;k) = 0 for 0< i < N. If X is a G- space, let Xtf = (Etf x X) IG ~ Xc. Then restriction induces an isomorphism

Hb(x; k) ----+ H i (xtf; k)

for all i < N. (See [72].)

1.3 The Localization Theorem

Before beginning this section in earnest, we need some re marks on neighbourhoods. Let G be a compact Lie group and let X be a G-space. Let x E X and suppose U ~ X is open with G(x) ~ U. Then, because of the compactness of G, it is easy to see that there is an open invariant V ~ X such that G(x) ~ V ~ U. Now suppose that A ~ X is invariant, U ~ X is open and A ~ U. Then, by taking unions over the orbits in A, there is an open invariant V ~ X such that A ~ V ~ U. If X is normal, e.g., paracompact, it is now easy to deduce that if A ~ X is closed and invariant, and if U ~ X is open with A ~ U, then there is an open invariant V ~ X such that A ~ V ~ V ~ U; and, of course, V is


also invariant. Now if A ~ X is closed and invariant, and if X is paracompact, then the tautness property of Alexander-Spanier cohomology implies that

He(A; k) ~ ~ He(V; k), ~

where the direct limit is taken over all invariant open neighbourhoods of A directed downward by inclusion. (Since X is paracompact, the direct limit can also be taken over all invariant closed neighbourhoods of A.) This follows from the tautness property of Alexander-Spanier cohomology in the non-equivariallt case together with Remarks 1.2.4(3).

Definition 1.3.1. Let X be a G-space and let xE X. Let jx : G(x)c ------7

Xc be the inclusion. In view of Remarks 1.2.3{3}, jx induces a homomorphism

j; : He(X; k) ------7 H*(BGx ; k).

When k is a field of characteristic 0, by Remarks 1.2.4(2}, if we restriet further to ~ i.e., consider the composition

j;o : He(X, k) ~ H*(BGx ; k) ------7 ~ k),

then ker(j;o) = ker(j;). If V ~ X is an invariant subspace, we shalt let jF denoted the in

clusion Vc ------7 Xc.

By tautness, if X is paracompact, and if x E X and if u E He (X; k), then j;(u) = 0 if and only if there is an invariant open neighbourhood V of G(x) such that ~ = O.

We can now prove the following importallt lemma of Quillell. (See [72], Proposition 3.2.) For simplicity we shall state alld prove the lemma assuming that X is compact: for a more general treatment see, e.g., [4], Proposition (3.2.1).

Lemma 1.3.2. Let G be a compact Lie group and let X be a compact G-space. Let u E He(X; k), where k is any commutative ring. Suppose that j; (u) = 0 for alt x EX. Then u is nilpotent: i. e., un = 0 for some positive integer n.


Proof Since j;(u) = O,jvx (u) = ° for some invariant open neighbourhood Vx of G(x), by tautness. Since X is compact, there are points Xl, ... ,Xn E X such that X is covered by VXI ' ••• , VXn • Let Vi = VXi •

Since jV, (u) = 0, there is Ui E HG (X, Vi; k) such that Ui restricts to t

n u. Now Ul ... U n E HG(X, U Vi; k) = 0; and Ul ... U n restricts to uno 0

i=l

Remark 1.3.3.

(1) If X is a manifold and G is acting smoothly, then the slices in X are linear and, hence, contractible. It follows that one can use tubes and slices instead of tautness in this case.

(2) If G = 1, the trivial group, and if U E H*(X; k) has positive degree, then j; (u) = ° for all X E X. Thus, if the lemma holds for all compact Lie group actions on X, every cohomology dass of positive degree must be nilpotent. This is a feature of finitistic spaces (i.e., those satisfying the Swan condition). And, indeed, Lemma 1.3.2 holds whenever X is paracompact and finitistic. (See [4], Proposition (3.2.1).) This generalization depends crucially on the result of [27J that X j G is finitistic if X is finitistic.

(3) The composition j;p* : H*(BG; k) ~ H*(BGx ; k) is simply the restriction homomorphism induced by the indusion of Gx into G, as is easily seen.

Definition 1.3.4. Let G be a torus of rank r. So H*(BG; Q) = Q[tl, ... , tr ], where deg(td = 2 for 1 ::; i ::; r. Let K <;;; G be a subtorus; and let jk : H*(BG; Q) ~ H*(BK; Q) be induced by the inclusion. Since G ~ K x G j K, jk is surjective. Let P K = ker(jk). Then P K is generated by homogeneous linear polynomials in tl, ... ,tr , for example, the images of a generating set for H*(B(G j K); Q) under the homomorphism induced by the projection G ~ G j K. (Note that H*(BG; Q) ~ H*(BK; Q) &;; H*(B(GjK); Q) by the Künneth theorem.)

Let p be any prime ideal in H* (B G; Q); and let O"p denote the prime ideal generated by p n H 2 (BG; Q). That is, O"p is generated by the homogeneous linear polynomials in p. It follows that O"p = P K for so me subtorus K <;;; G.

The Localization Theorem, which will be given below, usually involves localizing the R = H*(BG; ~ HG(X; k) with a multiplicative subset S <;;; R. (Note that, if M is a ~ if S ~ R is a


multiplicative subset, then the localized module, 8-1 M, is defined to be 8-1 R ® M. And, if R is an integral domain, then 8- 1 R is the subring of

R the field of fractions of R which consists of all fractions with denomina-tors in 8. See, e.g., [7], Chapter 3.) The following definition introduces some notation which is useful for stating the Localization Theorem.

Definition 1.3.5. Let G be a torus, let X be a G -space and let R = H*(BG; Q). Let f E Hc(X; Q), and let Xl = {x E X;j;(f) f- O}.

Now let 8 ~ R be a multiplicative set: i.e., 1 E 8, and ab E 8, whenever a E 8 and b E 8. Let

X S = {x E X;j;p*(s) f- 0 for all sE 8}.

So X S = n Xp*(s). sES

Since j; involves restriction to the orbit of x, clearly X land XS are invariant subspaces of X. By tautness, or tubes and slices in the smooth case, the complement X - X I is open: hence, X land X S are closed.

One defines X I and X S similarly when G is any compact Lie group and cohomology is taken with coefficients in any commutative ring.

If K ~ G is a subtorus, let 8(K) ~ R be the multiplicative set generated by the homogeneous linear polynomials not in P K. So 8 (K) consists of 1 and all products of homogeneous linear polynomials not in PK.

Lemma 1.3.6. Let G be a torus and K ~ G a subtorus. Let X be a G-space. Let p be a prime ideal in R = H*(BG;Q) with ap = PK. Let 8 ~ R be any one of the multiplicative sets R - p, R - PK or 8(K). Then X S = X K, the fixed point set of K acting on X.

Proof Let 81 = R - P and 82 = R - P K. From the definition of ap, 8(K) ~ 81 ~ 82 . So X S2 ~ XSI ~ XS(K).

Now, by Remarks 1.3.3(3) and 1.2.4(2), for x E X and s E R, j;p*(s) = 0 if and only if s E ~ And

~ ~~

Thus X S2 = X K ; and XS(K) ~ X K . 0

We shall now state and prove the Localization Theorem for compact G-spaces, where G is any compact Lie group; and we shall consider


multiplicative sets 8 <;;;; H*(BG; k), where k is any commutative ring. In this generality, to avoid unnecessary difficulties, one should assurne that 8 is celltral: i.e., as = sa, for any s E 8 and a E H*(BG; k). We shall discuss useful special cases and generalizations in some remarks following the proof.

Theorem 1.3.7. (The Localization Theorem) Let G be a compact Lie group and let X be a compact G -space. Let

8<;;;; H*(BG; k) be a central multiplicative set, where k is a commutative ring. Let 'P : X S ~ X be the inclusion. Then the localized restrietion homomorphism

is an isomorphism.

ProofFollowing [72], first supppose that X S = 0. So, for any x EX, there is s(x) E 8 such that j;p* (s(x)) = O. By tautness, or tubes and slices, there is an open invariant neighbourhood V(x) of G(x) such that ~ = 0 in Hc(V(x);k). Choose X1, ... ,Xn E X such that

VI = V(xt}, ... , Vn = V(xn ) cover X. Let s = s(xt}··· s(xn ). Then s E 8; and j;p* (s) = 0 for all x E X. Thus p* (s) is nilpotent by Lemma 1.3.2; and so 8- 1 Hc(X; k) = O.

Now suppose that X S =I- 0. Let V be a closed invariant neighbourhood of X S ; and let W be the complement of the interior of V. So WS = (V n W)s = 0.

An excellent property of Alexander-Spanier cohomology , at least for paracompact spaces, is that the Mayer- Vietoris sequence is exact for arbitrary pairs of closed subspaces: see [15], Chapter 11, Section 13. And, fortunately, if G is a compact Lie group and X is a paracompact G- space, then Xc is paracompact also: see [4], Section 3.2. Thus there is a Mayer-Vietoris sequence

... ~ Hb(X; k) ~ Hb(V; k) EB Hb(W; k) ~

Hb(vnW;k) ~ ...

Since localization is exact, by the first part of the proof,

8- 1 Hc(X; k) ~ 8-1 Hc(V; k)

is an isomorphism. The result now follows by tautness, since localization commutes with direct limits. 0


Remark 1.3.8.

(1) Consider <p* : Hc(X; k) -+ Hc(XS; k). The fact that S-l<p* is an isomorphism means the following two things.

(a) If u E ker(<p*) ~ Hc(X; k), then there is sES EUch that su = 0: i.e., u is S- torsional.

(b) For any v E Hc(X s ; k), there is sES such that sv E im(<p*).

(2) In these notes we shall concentrate on the case where G is a torus and k = Q. And, usually, S will be one of the multiplicative sets listed in Lemma 1.3.6. The interplay among the three types of multiplicative sets listed in Lemma 1.3.6 is very important for the general theory, see, e.g., [4], Section 3.7; but here we shall concentrate on the sets R-PK and S(K). Note that PG = (0) and S( G) is generated by all non-zero homogeneous linear polynomials in R = H*(BG; Q). In particular, to obtain information about the fixed point set, Xc, one can localize at (0); and so R becomes R(o), its field of fractions.

Note, too, that, since (Xc)c = BG X Xc,

a free R- module. In particular, one can expand upon re mark (l)(a), above, for <p* : Hc(X; Q) -+ Hc(Xc ; Q): U E ker(<p*) ~ Hc(X; Q) if and only if there is a nonzero s ERsuch that su = O.

Thus, for <p* : Hc(X; Q) -+ Hc(Xc ; Q), ker(<p*) is precisely the torsion submodule. And, by using S(G) instead of R - (0) , it follows that the annihilator (in R) of any torsional element of Hc(X; Q) contains a product of nonzero homogeneous linear polynomials in R. Similarly, given any v E Hc(Xc ; Q), there is a prod uct of homogeneous linear polynomials, sES ( G) , such that sv E im(<p*). We shall see a remarkable consequence of this fact in the next section: see Lemma 1.4.9 et seq.

(3) There are a number of ways to generalize Theorem 1.3.7. For example, if X is a paracompact finitistic G-space, then, with one extra condition, the theorem is valid as stated. The extra condition concerns the number of orbit types. In general, one must require that the set of conjugacy c1asses of isotropy subgroups, Gx , as x


ranges throughout X, be finite. If k is a field of characteristic 0, then it is enough that the set of conjugacy elasses of identity components of isotropy subgroups, ~ be finite. The latter, of course, always holds when G = 81, the drele. See [4], Section 3.2, for details.

It is often convenient, for other reasons, if the number of orbit types (Le., the number of conjugacy elasses of isotropy subgroups) is finite. So the following theorem of [61] is useful.

Theorem. 1f M is an orientable topological manifold, and if H*(M; Z) is jinitely generated, then any action of a compact Lie group on M has jinitely many orbit types.

(4) Another observation, which is useful for the general theory, is the following. Let G be a torus and let K <;;; G be a subtorus. Let R = H*(BG; Q) and let 8 <;;; R be one of the multiplicative subsets of Lemma 1.3.6: i.e., 8 = R - p, where ap = P K, or 8 = R - P K or S = S(K). Let is : HC(XK;Q) -+ 8- 1Hc(X K ;Q) be the localization homomorphism u t---+ u/1. Then is is injective.

This follows easily since (XK)c ~ BK x (XK)C/K; and so

HC(XK ; Q) ~ H*(BK; Q) l8l HC/K(XK ; Q).

(5) Localization destroys grading. But all the multiplicative sets, which we use, consist of elements of even degree only. So, after localization, one can still distinguish even from odd. For example, one can still consider Euler characteristics after localization: see Corollary 1.3.9, for example.

(6) The Localization Theorem, in this context, was first formulated by Hsiang and Quillen, independently ([50], [72]). It depends, however, very heavily on the pioneering work of Borel ([13]). Hsiang and Quillen were also inspired by similar results due to Atiyah and Segal in equivariant K - theory; and tom Dieck obtained localization theorems for other cohomology theories.

We shall now give some easy corollaries of the Localization Theorem.

Corollary 1.3.9. Let G be a torus and let X be a compact G - spacewith diIllQH*(X;Q) < 00. Then the Euler characteristics of X and XC are equal:


ProofLet E;'* be the Leray-Serre spectral sequence for Xc ----t BG in rational cohomology. Let R = H*(BG; Q). Since the rows are Rmodules, and the differentials, dr , r ~ 2, are R-module homomorphisms, one can localize the spectral sequence at (0), i.e., with S = R - (0). Let K = R(o), be the field of fractions. The spectral sequence, S-lEr ,

becomes a sequence of complexes of vector spaces over K. And S-l E2 ~ K ®Q H*(X; Q). Now, computing Euler characteristics over K, one has

since E oo = Er for so me finite r (because dimQH*(X;Q) < (0), and Euler characteristic is preserved when taking homology. But

x (S-l E oo ) = X (s-l He (X; Q) ) = X (S-l He(XC; Q)) = X{XG),

because He(XG;Q) ~ RQ!)H*(XG;Q). IQJ

The proof above shows that

Indeed, one has, with the notation of the proof above,

dimQ H*(X; Q) dimK(S-l E 2 ) 2:: dimK(S-l E3) 2:: ... ~ dimK(S-lEoo ) = dimK (S-lHe(X;Q)) = dimK (S-lHe(Xc ;Q)) = diillQH*(XG;Q).

Thus we have the next corollary.

o

Corollary 1.3.10. Let G be a torus and let X be a compact G-space with dimlQJ H* (X; Q) < 00. Then

diillQ H* (XG; Q) ~ diillQ H* (X; Q).

Furthermore, dimlQJ H* (XC; Q) = diillQ H* (X; Q) if and only if E 2 = E oo in the Leray-Serre spectral sequence for p : Xc ----t BG in rational cohomology (i. e., X is TNHZ in Xc ----t BG with respect to rational cohomology).

Remark 1.3.11. Because the distinction between even degrees and odd degrees survives localization (see Remarks 1.3.8(5)), the inequalities of Corollary 1.3.10 hold separately for even degree cohomology and odd degree cohomology .


Example 1.3.12. Let G = T r and let X be a compact G-space with H*(X; Q) ~ H*(cpn; Q). Let H*(X; Q) = Q[x]/(xn+1), where deg(x) = 2. Let R = H*(BG; Q) = Q[tl, ... , t r ]. Since X(X) = n + 1, XC =I 0 anri X(Xc ) = n + 1, by Corollary 1.3.1. Also, since Hodd(X; Q) = 0, Hodd(XC; Q) = 0, by Corollary 1.3.2 and Remark 1.3.9. And X is TNHZ in Xc -+ BG: see Remarks 1.2.4(1) . Thus diIIlQ H*(Xc ; Q) =

diIIlQ H*(X; Q) = n + 1.

Now consider the Leray- Serre spectral sequence for X -Ä. Xc -4 BG. Since i* is surjective (see Seetion 2), there is x E Hb(X; Q) such that i*(x) = x. Also, eaeh row, E;,2j = g;;}j, is a free R- module generated by x j (0 ::; j ::; n). And so, using the inereasing filtration, it follows that Hc(X; Q) is generated by x as aR-algebra. Indeed, sinee x n+1 = 0,

Hc(X; Q) ~ R[x]/ (f(x)) ,

where j(x) is a monie polynomial of degree n + 1 with eoefficients in R. There is only one relation sinee K[x] is a prineipal ideal domain, where K is the field of fraetions of R, and there is a standard argument using eontent (i.e., Gauss's Lemma). And the relation is monie of degree n+ 1 just beeause x n+1 = 0.

Now let F I , ... , Fs be the eomponents of Xc. Let cp : XC -+ X and Cpj : Fj -+ X be the inclusions. Sinee (XC)c ~ BG X Xc,

s

Hc(Xc ; Q) ~ EBR 0 H*(Fj; Q). j=l

Let cpi(x) = ai + Ui E H'2(BG; Q) EB H 2(Fi; Q) ~ Hb(Fi; Q), for 1 ::; i ::; s. Since xn+1 is aR- linear eombination of 1,x, . .. ,xn, via the relation j(x) = 0, u7+1 = 0, on aeeount of the bigrading in Hc(Fi; Q) =

R 0 H*(Fi; Q). (It is also clear that u7+ I = 0, beeause Ui = cpi(x) in ordinary, i.e., non-equivariant eohomology.)

Now let v E Hj (Fi; Q) ~ Hb(Fi; Q) ~ Hb(XC; Q), where j > o. By the Loealization Theorem, there is a nonzero r E Rand y E Hc(X; Q) such that cp*(y) = rv. (See Remarks 1.3.8(1)(b).) Sinee y is aR-linear eombination of 1, x, ... , xn, it follows that v is a rational multiple of apower of Ui. Thus H* (Fi; Q) is generated by Ui E H 2 (Fi; Q). So H*(P·rr1\) ~ H*(cpmi .rr1\) for some m· ° < m· < n sinee un+1 - ° t, ~ - ~ t, _ t -, i -.

(As many standard examples show, Fi may be just an isolated point: so mi might be zero, whieh happens if Ui = 0.) Thus cpi ((x - admi+1) =


Umi+1 = Ü· and so ~ ,

Since Hc(X; Q) is a free R-module, <p* is injective, by Remarks 1.3.8(I)(a). s s

So I1 (x-ad mi+l = ü. But the degree ofthis polynomial is L (mi+l) = i=l i=l

s

L dimQH*(Fi;Q) = dilliQH*(Xc;Q) = dimQH*(X;Q) = n+1. Thus, i=l

s

j(x) = II (x - admi+l .

i=l

Furt hermore , al, ... , as are distinct. To see this, write elements of Hc(XC; Q) in the form (Zl, ... , zs) where Zi E Hc(Fi ; Q) for 1 ~ i ~ s. Let ei be the element with Zi = 1 and Zj = Ü for j i= i. By the Localization Theorem, there is r E Rand y E Hc(X; Q) such that <p*(y) = rei. Since <p* (x) = (al +UI, ... ,as+us)' and for any bE R, <p*(b) = (b, . .. ,b), it follows that, for a polynomial g(x) E R[x] , the part of <p* (g(x)) in R EB ... EB R, is (g(ad, ... ,g(as)). So, since y = g(x) for so me g, it follows that aj i= ai, whenever j i= i.

s

Note that, because L (mi + 1) = n + 1, mi < n for all i, 1 ~ i ~ s, i=l

unless XC is connected.

Remark 1.3.13. In Example 1.3.12 one obtains a presentation of the algebra Hc(X; Q) in which the ideal of relations contains information ab out the fixed point set. Hsiang's Fundamental Fixed Point Theorem is a significant generalization of this: see [51], Theorem (IV.I), or [4J Section 3.8.

Another generalization of Example 1.3.12 concerns the number of generators and relations for H*(F; Q), where F is a component of Xc, compared to the corresponding numbers for H*(X; Q): see, for example, [4], Theorem (3.8.12).

A third, and straightforward, generalization of Example 1.3.12 is as follows. Suppose that G is a torus, X is a compact G-space, and H*(X; Q) is generated by elements of degree 2. Then X is TNHZ in Xc ----+ BG, and XC i= 0. Let F be a component of Xc. Then F is acyclic over the rationals, or H*(F; Q) is generated by elements of


degree 2. This is useful, especially when combined with the result on the numbers of generators and relations mentioned above, because many interesting spaces have their rational cohomology generated by elements of degree ~ simplicial toric varieties, for example.

An easy consequence of the Localization Theorem is the following. If G is a torus and X is a compact ~ then XC 1= 0 if and only if p* : H*(BG;Q) ---t HG(X;Q) is injective. If there is a fixed point, x, say, then p : Xc ---t BG has a section given by [z] f-----+ [z, x]. Conversely, if p* is injective, then ~ is a free module; and so HG(X; Q)(O) ,

the localization at (0) ~ H*(BG; Q), is not zero: hence XC 1= 0. Hsiang generalized this; and a very substantial furt her generalization was carried out in [19]; see, e.g., [4], Section 3.7. Hsiang's result is the following. (Recall that, for an ideal J ~ R, .JJ = {a E R; an E J, for some integer n 2 I}, the radical of J. See, e.g., [7], Chapter 1.)

Proposition 1.3.14. Let G be a torus and let X be a compact G-space. Let K 1 , ... , K n be the identity components oJ the maximal isotropy subgroups. Let J = ker(p* : H*(BG;Q) ---t HG(X;Q)]. Then .JJ = n

n PKi· i=l

(See Definitions 1.3.4 for the definition of P K.) Proof By the existence of tubes and slices, for any x E X, there is

an open neighbourhood V(x) of x such that, for any y E V(x), Gy ~ Gx :

see [14], Chapter II, Corollary 5.5. Since X is compact, a finite number of V(x)s cover X; and so there are finitely many maximal isotropy subgroups. Also .JJ = P1 n·· · n Pm, where P1 , ... ,Pm are the smallest prime ideals containing J : see [7], Lemmas 7.11 and 7.12. So it remains to show that each Pi = P K i for so me subtorus K i , and that each such K i is maximal among the subtori K with X K 1= 0.

Let p be a prime ideal such that J ~ p. Then the localization at p, HG(X; Q)p, is not zero, since 1/1 1= O. Hence X K 1= 0, where crp = PK, by the Localization Theorem and Remarks 1.3.8(4). Hence HG (X; Q) P K 1= 0; and so J ~ P K. Thus, each minimal prime of J, Pi, 1 ::; i ::; m , has the form P Ki for a sub torus K i with X Ki 1= 0. The result now follows, because the correspondence K ---t P K is order reversing, and because X K = 0 if J is not contained in P K (for then 1/1 = 0 when localizing at PK). D


1.4 Poincare Duality

In this section, we shall begin by stating, without proof, some of the main results concerning Poincare duality and torus actions. And then we shall state and prove the Topological Splitting Principle and the Borel Formula.

Definition 1.4.1. Let X be a connected space with

dimQH*(X;Q) < 00.

Suppose that there is a non- negative integer n such that

Hi(X; Q) = 0 for all i > n;

and suppose that Hn(X; Q) ~ Q. Suppose, too, that, for 0 ~ i ~ n, and non-zero x E Hi(X, Q) , there is y E Hn-i(X; Q) such that xy i= O. Then X is said to satisfy the Poincare d'uality over the rationals, or X is said to be a rational Poincare duality space; and n is said to be the formal dimension of X: one writes fd(X) == n.

The following theorem is due to Bredon alld Chang and Skjelbred. For the original proofs see [16] and [20]. For other proofs see [4], Section 5.2, or [73]. These references also give the other conditions under which the theorem holds.

Theorem 1.4.2. Let G be a torus and let X be a compact G - space. Suppose that X is a rational Poincare duality space, and that XC i= 0. Let F be a component of Xc . Then F is a rational Poincare duality space.

Furthermore, fd(F) ~ fd(X) and fd(X)-fd(F) is even. And fd(F) = fd(X) if and only if XC is connected and the restriction homomorphism H*(X; Q) -+ H*(F; Q) is an isomorphism.

Now let X and Y be G- spaces, where G is a torus, and let f : X -+ Y be an equivariant map. Suppose that X alld Y are rational Poincare duality spaces. Then it is ~ to define an equivariant Gysinor umkehr homomorphism f? : Hc(X; Q) -+ Hc(Y; Q) of degree fd(Y) - fd(X) : Le., for any

u E Hb(X;Q) , f!c(u) E Hb+n-m(Y;Q) ,

where fd(X) = m and fd(Y) = n. See [4], Section 5.3, for the details of the definition and for the proof of the following properties.

1.4. Poincare Duality 21

Theorem 1.4.3. Let G be a torus, let X, Y and Z be compact G-spaces which are also rational Poincare duality spaces. Let f : X -t Y and g : Y -t Z be equivariant maps. Then

(1) f!G : Hc(X; Ql) -t Hc(Y; Ql) is a homomorphism of H*(BG; Ql)modules;

(2) (gfW = gr f!G;

(3) (lx W is the identity on Hc(X; Ql), where Ix X -t X is the identity on X;

(4) for any x E Hc(X;Ql) and y E Hc(Y;Ql),

f? (xj*(y)) = f?(x)y;

(5) if G = 1, the trivial group, then H is the famiZ,iar Gysin (or umkehr) homomorphism defined in terms of Poincare duality and the dual of 1*; and

(6) if K ~ G is a subtonls, then the following diagram commutes

Hc(X; Ql) f.o

Hc(Y; Ql) ~

1 1 HK(X; Ql)

f/( Hj((Y;Q). ~

where the vertical homomorphisms are induced by the standard maps XK -t X G and YK -t YG and the horizontal maps are the respective equivariant Gysin homomorphisms.

We shall apply the equivariant Gysin homomorphism to the inclusion of a component of the fixed point set of a G-space, X, into X. We shall use the following assumptions and notation for the rest of this section.

Assumptions : Let G be a torus, and let X be a compact G-space. Suppose that X is a rational Poincare duality space of formal dimension n. Suppose that X G i- 0, and let F I , ... ~ be the components of XG. Let fd(Fd = ri for 1 ::; i ::; s. And let 'P : X G -t X and 'Pi : Fi -t

X, for 1 ::; i ::; s, be the inclusions. We shall choose generators, i.e., top classes, (X) E Hn(X;Ql) and (Fi ) E HTi(Fi;Ql) for 1 ::; i ::; S; hence 'P{! ((Fi )) = (X), where 'P{! is the ordinary (non-equivariant) Gysin


homomorphism as in Theorem 1.4.3(5). (As usual - see [4], Section 5.3 - the definition of f? : Hc(X; Q) ----+ Hc(Y; Q) depends upon a choice of top dasses, corresponding to a choice of orientations in the case of smooth manifolds.)

Definition 1.4.4. Under the above assumptions, Jor 1 ::; j ::; s,

(1) letBj = ~ E ~ where 1 E H&(Fj;Q) is the identity;

(2) let Uj ~ E Hä(X; Q), vzewzng Hn(Fj ; Q) ~ Hä(Fj ; Q);

(3) let ej = <pj(Bj ) E ~ and

(4) let nj be the part oJ ej in Hn-rj (BG; Q), using the isomorphism Hc(Fj ; Q) ~ H*(BG; Q) ® H*(Fj ; Q).

The dass ej is called the equivariant Euler dass oJ Fj .

Proposition 1.4.5. With the notation and assumptions above,

(1) Jor any u E Hc(X; Q), ~ <pj(u) = Bju;

(2) for any v E Hc(Fj ; Q), ~ = ~ = ejv; in particular, ~ = 0 iJi i= j; and ej = <p*(Bj );

(3) i*(Uj) = (X), where i: X ----+ Xc is the indusion oJthejibre; and

Proof (1) ~ = ~ (l<Pj(u)) = ~ by Theorem 1.4.3(4). And

~ = Bj , by definition. (2) By the Localization Theorem, there is non-zero

a E H*(BG; Q) and u E Hc(X; Q) such that <p*(u) = av. Thus ~ = ~ = <p*(Bju) = a<p*(Bj)v.

So ~ = <p*(Bj)v, since Hc(XG ; Q) is a free module. But <p*(Bj)v = <pj(Bj)v, since v E Hc(Fj; Q). And <pj(Bj ) = ej, by definition.

(3) This follows at once from Theorem 1.4.3(5) and (6), with K = 1, since <p} ( (Fj)) = (X).

(4) <p*(Uj) = ~ ((Fj)) = ej(Fj), by (2); and ej(Fj) = nj(Fj), by definition of nj , since (Fj ) is the top dass. 0


Definition 1.4.6. For v E HC(X C ; Q), let

I(v) = {a E H*(BG;Q);av E im(<p*)}.

Clearly I(v) is an ideal. And I(v) :I (0), by the Localization Theorem.

Lemma 1.4.7. I((Fj )) = (Oj), the principal ideal generated by Oj.

Proof Oj E I((Fj )) by Proposition 1.4.5(4). Let a E I((Fj )). So there is U E Hc(X; Q) such that <p*(u) = a(Fj ). Thus <p*(aUj-Oju) = O. So aUj - OjU is torsional, by the Localization Theorem.

Now let I: Hc(X; Q) ----+ H*(BG; Q) = R, be the R-module homomorphism defined as folIows. Recall the fd(X) = n. In the Leray-Serre spectral sequence for Xc ----+ BG, ~ = E;,n, by Proposition 1.4.5 (3). And E;,n is a free R-module generated by (X). Using the increasing filtration described in Section 1.2, there is a quotient homomorphism

7rn : Hc(X;Q) = PHc(X;Q) ----+ PHc(X;Q)/P-1 Hc(X;Q).

Note that F n Hc(X; Q)j:p,-l Hc(X; Q) = ~ = E;,n. Define I for y E Hc(X; Q), by 7rn (Y) = I(y)(X). Since aUj - OjU is torsional and E;,n is free, I(aUj - Oju) = O. And

I(Uj) = 1, by Proposition 1.4.5(3). So a = OjI(u). 0

Lemma 1.4.8. Oj is a product of homogeneous linear polynomials in H 2(BG; Q).

Proof By Remarks 1.3.8(2), I( (Fj)) contains a product of homoge-neous linear polynomials; and Oj is a factor of this product. 0

Definition 1.4.9. To simplify the notation, let Fj = F,Oj = 0, Uj = U, and so on. Let 0 = ~ W-:;'2 ... W;:k, where Wl, ... , Wk are the distinct homogeneous linear factors of 0 and ml,. , , , mk are their multiplicities. (So k,Wl, ... ,Wk and ml, ... ,mk depend on j.) Now, by Definition 1.3.4, the principal ideal (wd = PKi, for some subtorus K i ~ G of corank one, for 1 ::; i ::; k.

For any subtorus L ~ G, let F(L) be the component of XL which contains F. (More precisely, let Fi(L) be the component of XL which contains Fi .)

Theorem 1.4.10. With the notation above, if L ~ G is a subtorus, then fd (F(L)) > fd(F) if and only if L ~ Ki; for some i, 1 ::; i ::; k.

Furthermore, fd (F(Kd) - fd(F) = 2mi for 1 ::; i ::; k.

24 1. Loealization Theorem and Symplectie Torus Actions

Before proving the theorem we shall state two lemmas. For the first lemma, we suppose that G is a torus, X and Y are compact G-spaces, X and Y are rational Poincare duality spaces, J : X ----t Y is an equivariant map, and a subtorus K ~ G is acting triviallyon X and Y. ThllS, for example, HG(X; Q) = H*(BK; Q) @ Hi(X; Q), where L = G / K. The proof of thc first lemma, which will not be given, follows flom the definition of the equivariant Gysin homomorphism. (See also [4], Proposition (5.3.3)(3).)

Lemma 1.4.11. Let G, X, Y, J, K and L be as in the paragraph above. In partieular, the sub torus K acts triviallyon X and Y, and L = G / K. Then J? = 1 0 J!L, where 1 is the identity on H* (B K; Q).

For the second lemma, return to the situation of Theorem 1.4.10: thus G is a torus, X is a cOlnpact G-space and rational Poincare duality space, XC # 0, Fj is a component of Xc, and, for a subtorus L ~ G, Fj(L) is the componellt of XL which contains Fj. Let 'ljJ : XL ----t X and 'ljJj : Fj(L) ----t X be the inclusions. Let fd(X) = n and fd (Fj(L)) = nj(L).

Then ~ E ~ (Fj(L);Q). Let ~ = ej(L,G). Let

ej(L) E HZ-nj(L) (Fj(L); Q) be the equivariant Euler class of Fj(L) with respect to the action of L on X. So ej(L) = ~ = ~ by

Proposition 1.4.5(2). Finally, let Oj(L) = ~ E ~ Q).

Lemma 1.4.12. With the eonditions and notation of the paragraph above

(1) for any v E HG (Fj(L); Q) ~ = ~ = ej(L, G)v; in partieular, ej(L, G) = ~ and

(2) under the restrietion homomorphism

rL : HG( ; Q) ----t Hi( ; Q), ej(L, G)

goes to ej(L) : i.e., rL (ej(L, G)) = ej(L).

Proof As in the proof of Proposition 1.4.5(2), for

v E HG (Fj(L); Q) ,

there is a E R - PL where R = H*(BG;Q) and u E HG(X,Q) such that 'ljJ*(u) = av. Thus

~ = ~ = ~ = 'ljJ* (Oj(L)u) = aej(L,G)v,


where we used Theorem 1.4.3(4) for the third equality. (1) now folIows, since He(X L; Q) has no (R - PL)-torsion: see Remarks 1.3.8(4).

To see (2), consider the following diagram, which commutes by Theorem 1.4.3(6).

He(Fj(L); Q) 7/Jc, -4 He(X;Q)

ril lri -----7

Hj)Fj(L); Q) ~ Hi(X; Q).

SO ~ = ~ = ~ Hence

rL(ej(L, G)) = rt'ljJ*'ljJJ(l) = 'ljJ*rt'ljJJ(l) = 'ljJ*'ljJ/(1) = ej(L).

o Proof of Theorem 1.4.10 We shall use the notation established above.

So for a subtorus L ~ G, 'ljJ : XL -----7 X and 'ljJj : F(L) = Fj(L) -----7 X are the inclusions. (The component F of XC is Fj for some j, 1 :s: j :s: s.) Let H*(BL; Q) = H'i, and let He = R. Let .\ : F -----7 F(L) be the inclusion, and recall that <p : XC -----7 X and <Pj : F -----7 X are the inclusions. 0

Also let fd(X) = n, fd(F) = r, fd (F(L)) = n(L), and, for 1 :s: i :s: k, fd (F(Kd) = ni.

Now the equivariant Euler class of F, ej, satisfies

ej = ~ = ~ = ~ = .\* (ej(L,G).\((l)) , by Lemma 1.4.12(1).

For any v E He(F; Q) ~ R 0 H*(F; Q), let vo be the component of v in R 0 HO(F; Q) ~ R. Thus, by definition of n = nj , n = (ej)o = .\* (ej(L, G))o.\*.\( (1)0; and .\* .\((1) E HT(F; Q), where T = G / L, ~

Lemma 1.4.11. In particular, .\* .\((1)0 E H;(L)-r.

Recalling that the generators of Hr are also generators for the ideal PL (see Definitions 1.3.4), it follows that nE PL ifn(L) > r. Thus, if n(L) > r', some Wi E PL, 1 :s: i :s: k; and so PKi ~ PL : hence L ~ Ki.

Since L ~ K i implies that F(Kd ~ F(L), to finish the proof, it is enough to show tImt ni - r = 2mi for 1 :s: i :s: k.

In the calculations above with L = K i , since HC/K • = Q[w;] , .\*.\((1)0 = O'Wfi, where 0' E Q,O' i- 0, and Ci = ~ - r). Since n = .\* (ej(Ki, G))o.\* .\((1)0. to complete the proof, it is enough to


show that Wi is not a factor of ). * (ej (Ki , G))o. More gene rally, it is enough to show that ).* (ej(L, G))o rf:. PL.

For x E F, and indusion i : {x} -+ F, identifying HG( {x}; Q) with R, it is enough to show that i*)'* (ej(L, G)) rf:. PL; or, with r'i as above (see Lemma 1.4.12(2)), it is enough to show that r'ii*)'* (ej(L, G)) i- O. But

rii*)'* (ej(L, G)) = ()'i)*r'i (ej(L, G)) = ()'i)* (ej(L)).

Abbreviating ()'i)* (ej(L)) = ~ ~ is precisely the dass corresponding to o for F(L) <:: XL and the action of L on X. Thus

{a E Hl;a(F(L)) E im[1/;* : Hl(X;Q) -+ Hl(X L;<Q))]} = (0,

by Lemma 1.4.7; and so ~ i- 0 by the Localization Theorem. D

With X, Fand F(L) above, and n = fd(X), r = fd(F) and n(L) = fd (F(L)) , one has immediately the Borel Formula ([13], Chapter XIII, Theorem 4.3), namely the following.

Corollary 1.4.13. We have n - r = l: (n(L) - r), where the sum on L

the right is over all subtori, L, having corank one.

k Proof By Theorem 1.4.10, l: (n(L) - r) = l: (ni - r), where ni =

L i=l n(Kd. The result follows since

k k

n - r = deg(O) = ~ .. . W;;k) = L2mi = L(ni - r). i=1 i=1

D

Remark 1.4.14.

(1) Lemma 1.4.8 and Theorem 1.4.10 combined are known as the Topological Splitting Principle.

If X is a smooth manifold and if the torus, G, is acting smoothly, then, for any x E F <:: X G , G acts linearly on the tangent space Tx(X). If one chooses an invariant Riemannian metric on X, then the representation Tx(X) splits as Tx(F) tB Nx(F), where Nx(F) is the normal space to F at x, and the tangent space of F at x, Tx(F), is the trivial part of the representation. Now the real representation of the torus G, Nx(F), splits as a sum of non-trivial twodimensional representations ml VI tB ... tB mk Vk , where VI,· .. , Vk

1.4. Poincare Duality. 27

are the distinet two-dimensional representations, and ml, ... , mk k

aretheirmultiplicities. Inparticular, 2: 2mi = dimNx(F) = n-r, i=1

where n = dim(X) and r = dim(F). And eaeh Vi is determined by a homomorphism Pi : G --t SI, for whieh the identity eomponent of the kernel, ker(pi)O = K i, say, is a eorank one subtorus. It is remarkable that the Topologieal Splitting Principle recovers so mueh of this information assuming only that X satisfies Poineare duality over the rationals.

(2) Suppose that M is a closed orientable topologieal manifold of dimension 2n; and suppose that the n-torus G = T n is aeting effeetively with MG i- 0. It follows at onee from the Borel Formula that fd(F) = ° for any eomponent F ~ MG. From [13], Chapter V, Theorem 3.2 (due to Conner and Floyd), one has that Fis an orientable eompaet integral eohomology manifold. Sinee fd(F) = 0, F is an isolated point. ([13], Chapter I, Corollary 4.6.) So MG is finite.

Example 1.4.15. Consider Example 1.3.12. Let

s

Vi = (x - ai)mi II (x - aj )mj+l. j=I,ji:i

Then i*(Vi) = x n , the top dass of X (where, as usual, i : X --t XG is the inclusion of the fibre). And

s

s

cp*(Vi) = ur;i II (ai - aj)mj+l. j=I,ji:i

Thus Oi = TI (ai - aj )mj+l. So there are s - 1 eorank one subtori, j=I,ji:i

K, with fd(Fi(K)) > fd(Fi ); and they are determined by PKj = (aiaj),l ::; j ::; s,j i- i. Furthermore, fd (Fi(Kj)) - fd(Fd = 2(mj + 1) for 1 ::; j ::; s, j i- i. (This gives another proof that ai i- aj for i i= j.)

Remark 1.4.16. All the results of this seetion remain valid for paraeompaet finitistie G-spaees under the eondition deseribed in Remarks 1.3.8(3).

28 1. Localization The01'em and Symplectic Torus Actions

1.5 A Brief Summary of Symplectic Torus Actions

In this section we shall give the definitions of a symplectic manifold, a symplectic compact Lie group action and a Hamiltonian compact Lie group action with emphasis on the torus case. We shall state so me of the basic results including some of the fundamental results of Frankel [30J and their consequences. We shall end the section with the recent result of Jones and Rawnsley [53], the proof of which depends in a large part on Frankel's analysis. As before, we shall simplify matters by considering only the compact case: in particular, we shall only consider closed manifolds (except in a few general definitions).

Definition 1.5.1. Let M be a smooth manifold.

(1) If w is a p - form on M and V is a vector field on M, then i,,(w) zs the (p - 1) - form on M defincd by

iv(w)(VI , ... , Vp-d = w(V, VI,···, Vp- 1),

for any vector fields VI, ... , Vp- l '

(2) A 2-form, w, on M is said to be nondegenerate if, for any x E M, and any vector field V on M, iv( w) = 0 at x i1 and only if V = 0 at x.

(3) A closed nondegenerate 2- form on M is called a symplectic form on M. And, if there is a symplectic form, w, on M, then M , or, more precisely, (M, w), is said to be a symplectic manifold.

If a manifold, M, has a symplectic form, there are some immediate easy consequences as follows. (See, e.g. , [9] , Chapter II, Sections 1.2- 1.5, for more details.)

(1) dim(M) is even; (2) if dim(M) = 271, then wn is nonzero everywhere; (3) M is orientable and wn is a volume form on M ; (4) if M is closed, then wn is not exact, and, hence w is not exact;

and (5) M has an almost-complex structure.

Definition 1.5.2. Let (M,w) be a symplectic manifold.

(1) An action of a compact Lie group, G, on M is said to be symplectic if g*(w) = w, for all gE G.

1.5. A Brief Summary of Symplectic Torus Actions 29

(2) If f : M ----+ R is a smooth function on M, then, since w is nondegenerate , there is a vector field X I on M such that ix f (w) = df. XI is calted the symplectic gradient of f.

(3) If J, g : M ----+ Rare two smooth functions on M, then the Poisson bracket, {j, g}, of J and g is defined by

For a proof of the following proposition, see, for example, [9], Chapter n, Proposition 2.2.3.

Proposition 1.5.3. If (M, w) is a symplectic manifold, then the Poisson bracket makes COO(M) , the algebra of smooth real-valued functions on M, into a Lie algebra.

We shall now give the rat her complicated definition of a Hamiltonian compact connected Lie group action on a symplectic manifold. As we shall show after the general definition, however, in the case of a circle or torus action, the definition is much simpler: indeed it is very pleasing for a circle action.

Definition 1.5.4. Let (AI, w) be a connected symplectic manifold.

(1) Let V be a vector field on M. Then V is said to be locally Hamiltoni an if iv(w) is closed; and V is said to be Hamiltonian if iv(w) is exact.

(2) Let 1ie(M) be the real vector space of locally Hamiltonian vector fields on M, and let 1i(M) be the subspace of Hamiltonian vector fields. Let i : 1i(M) ----+ 1ic(M) be the inclusion. (Note that coker (i) ~ H 1 (M; R).)

(3) Let.'3 : COO(M) ----+ 1i(M) be defined by s(f) = XI, the symplectic gradient of J, for alt f E COO(M).

(4) Let G be a compact connected Lie group with Lie algebra g. Suppose that G is acting on M symplectically. For any a E g, let V(a) be the vector field on M defined at any x E M by

d V(a)(x) = dt [exp(ta)x]lt=o.

30 1. Loealization Theorem and Sympleetie Torus Aetions

Here exp : 9 ---+ G is the exponential map.

V(a) ean also be defined as follows. For any xE M, let 'Px : G ---+

M be the map 'Px (g) = gx. Then, at the identity 1 E G, 'Px induces a linear map of tangent spaees d'Px,1 : 9 = T l (G) ---+ Tx(M). And V(a)(x) = d'Px,1 (a).

Since the action is symplectie, g*(w) = w, for all 9 E G. This is equivalent to the eondition that .cV(a)(w) = 0, for all a E g, where, for a veetor field V, .cv(w) is the Lie derivative of w with respect to V. However, Cartan's formula gives

.cV(w) = ivd(w) + div(w)

(see, e.g., [10), Seetion 4.7, Problem 19); and so, sinee w is closed, .cv(w) = div(w). Thus .cv(w) = 0 if and only if V E lle(M). In partieular, the action of G on M is symplectic if and only if a I-----T V(a) is a function V : 9 ---+ lle(M).

(5) Let G be a eompaet connected Lie group with Lie algebra g. Suppose that G is acting sympleetically on M. Then the action is said to be Hamiltonian if there is a homomorphism of Lie algebras ji : 9 ---+ COO(M) such that isji = V:

COO(M) (Ji 9

sl 1V

ll(M) i

lle(M). ---+

Remark 1.5.5.

(1) Since the Lie algebra of SI, the circle group, is one dimensional, one only needs to consider a single associated vector field. That is, if G = SI is acting smoothly on a closed smooth manifold, M, then the associated vector field, V, is defined by

V(x) = ! [exp (21fit)x] It=o,

for all x E M.

If M is symplectic, with symplectic form w, then one has the following very simple characterizations.


(a) The action is symplectic if and only if iv(w) is closed. (See Definitions 1.5.4(4).)

(b) The action is Hamiltonian if and only if iv(w) is exact.

If the action is Hamiltonian then iv (w) = dh for some h E Coo (M); and h is called the moment map.

More generally, given a Hamiltonian action of a compact connected Lie group, G, on (M, w), the moment map is the function

p, : M --+ g* = Hom(g, R),

defined by p,(x)(a) = j:i(a)(x) , for all a E 9 and x E M, where j:i: 9 --+ COO(M) is as in Definitions 1.5.4(5).

(2) If G = SI X ••. X SI = TT, the r-torus, then an action of G on a symplectic manifold is Hamiltonian if and only if the actions of each of the factor circles is Hamiltonian, and, hence, if and only if the action of every subcircle is Hamiltonian. Indeed, it is not hard to show that this generalizes to any compact connected Lie group G: i.e .. the action of G is Hamiltonian if and only if the action of every subcircle is Hamiltonian; and it is sufficient that the action of at least one maximal torus is Hamiltonian. Cf. Remarks 1.5.7(1) below.

Let G = SI, and suppose that G is acting smoothly on a closed manifold M. There is a very interesting way to compute Hc(M; R) known as the Cartan model. Let 0inv be the subcomplex of the de Rham complex of M consisting of invariant forms: i.e., a differential form () is in 0inv if L:v(()) = 0, where V is the associated vector field of the circle action. (See Remarks 1.5.5(1) and Definitions 1.5.4(4).) Let t be an indeterminate of degree 2, and form the polynomial ring 0inv[t]. Define a derivation, D, on 0inv[t] by setting D(t) = 0 and D(()) = d() + iv{())t, for any () E 0inv' where d is the ordinary de Rham differential. Then D2 = 0; and Hc(M; R) = H(Oinv[t], D).

The R[t]-module structure is the same as the H*(BG; R)-module structure. Also, for i : M --+ Me the inclusion of the fibre in Me --+ BG, i* : Hc(M; R) --+ H*(M, R) is induced by the homomorphism 0inv[t] --+ 0inv given by evaluating at t = O. See [6] for more details.


Proposition 1.5.6. Let (M, w) be a closed symplectic mani]old with a symplectic action 0] the circle group G = SI. Let w = [w] E H 2(M; R). Then the action is Hamiltonian i] and only i]

w E im [i* : Hb(M; R) -t H 2 (M; R)] .

Proof If w E im [i* : Hb(M; R) -t H 2 (M; R)] , then there is an invariant 1- form 0 and an invariant function h E COO(M) such that D(w + dO + ht) = O. So i\l{w) = -dh - i\ldO = d{ -h + i\lO). (See [9], Chapter V, Excercise 2.2.2 and Proposition 3.1.1.)

Conversely, if i\l{w) = dh, then D(w - ht) = 0; and so w E im i* . 0

Remark 1.5.7.

(1) Proposition 1.5.6 is valid when G is any compact connected Lie group. A proof follows directly from the general form of the Cartan model, which is described in detail in [39]. See also [9], Chapter V, Proposition 3.1.1.

(2) Let G = SI and let (M, w) be a closed symplectic manifold with a Hamiltonian action of G. Let h : M -t R be the moment map. So idw) = dh. Since w is a nondegenerate, dh = 0 at x E M if and only if V(x) = o. Thus dh = 0 at x E M if and only if x E MG, the fixed point set. And, since M is compact, if the action is nontrivial, then h must have a maximum and minimum distinct from one another; and so MG must be nonempty with at least two components.

The main result of Frankel's paper [30] is that, for a Hamiltonian circle action on a closed symplectic manifold (M, w), the moment map h : M -t R is a Morse- Bott function. This means that given any fixed point, x, lying in a component F of the fixed point set, the Hessian of h (i.e., the symmetrie matrix of second order partial derivatives) is a nondegenerate symmetrie bilinear form on any subspace of the tangent space Tx{M) which is complementary to Tx(F).

The fact that h is a Morse- Bott function brings into play aB the immensely powerful geometry of Morse-Bott theory. In particular, associated with each component, F, of the fixed point set there is an index, which is defined to be the maximal dimension of a subspace of Tx{M), for x E F, on which the Hessian of h is negative definite. (This does not depend on the choice of x E F). One of Frankel's results is that these


indices are all even. We shall state another major result of [30] after the following definition.

Definition 1.5.8. Let X be aspace, let k be a field, and for each i 2: 0, let bi(X) = dimk Hi(X; k), the i-th Betti number of X over k. Let z be an indeterminate. Then the Poincare series (or polynomial) of X over k, P(X, z), is defined by

00

P(X,z) = I:bj(X)zj. j=O

P(X, z) is defined only if all bi(X) are finite; and it is a polynomial if all but finitely many bi(X) are zero.

For example, if G = TT ~ the r-torus, then, with any field of coefficients, P(BG,z) = (1- Z2)-T.

Now Frankel's result is the following.

Theorem 1.5.9. Let M be a closed symplectic manifold, let G = SI, the circle group, and suppose that G is acting on M in a Hamiltonian way. Let F1, •.• ,Fs be the components of MG. Let dim(M) = 2n, let dim(Fi ) = 2ri, and let the index of Fi be 2ki, for 1 ::; i ::; s.

Then, with coefficients in Q or in Fp , for any prime number p,

s

P(M, z) = I:>2ki P(Fi , z). i=l

Reversing the circle action, or replacing the symplectic form w by -w, one also has

s

P(M, z) = I: z2n- 2Ti- 2k i P(Fi, z). i=l

An important corollary is the following.

Corollary 1.5.10. Let M be a closed symplectic manifold, let G = TT, the r - torus; and suppose that G is acting on M in a Hamiltonian way. Then

(1) i* : Hc(M; Q) --t H*(M; Q) is surjective, where i : M --t MG is the inclusion of the fibre in the bundle p : MG --t BG;


(2) <p* : Hc(M; Q) ---+ Hc(MG; Q) is injective, where <p : MG ---+ M is the inclusion;

(3) Hc(M; Q) is a ]ree H*(BG; Q)-module; and

(4) the Serre spectral sequence ]or p : MG ---+ BG in rational cohomology collapses (i.e., E2 = Eoo).

ProofSince there are finitely many orbit types (see Remarks 1.3.8(3)), there is a subcircle K <;: G such that M K = MG. Since the action of K is also Hamiltonian (see Remarks 1.5.5(2)), applying Frankel's theorem with z = 1, we have

diIllQ H*(M; Q) = dimlQ H*(MG; Q).

Part (4) now follows by Corollary 1.3.10; and (1) is a standard consequence of (4) and the edge effect. (See the discussion following Remarks 1.2.3.) Part (3) is another standard consequence of (4), namely, the Leray-Hirsch Theorem ([76], Chapter 5, Sec. 7, Theorem 9). Part (2) follows immediately from (3) and the Loealization Theorem. (See Remarks 1.3.8(l)(a).) 0

Remark 1.5.11. For torus actions on compact spaces in general, for the parts of Corollary 1.5.10, (I)-{::=:} (3)-{::=:} (4) =* (2); but (2) does not necessarily imply (1), (3) and (4), unless G = SI. And there are plenty of examples where none of (1), (2), (3) or (4) hold.

Remark 1.5.12. Let (M,w) be a closed sympleetic manifold, let G = SI, and suppose that G is acting on M in a symplectic way. Suppose that MG f- 0. Let F be a eomponent of MG. Then F is also symplectie, and w restriets to a sympleetic form on F. This is not obvious, but it is a fairly easy consequenee of Frankel's geometrie analysis.

We finish this seetion with the Jones-Rawnsley Theorem, which is, in part, another consequenee of Frankel's Theorem 1.5.9. We shall indieate the proof briefly.

Theorem 1.5.13. ([53]) Let M be a closed symplectic manifold which admits a Hamiltonian action of G = SI, such that lv[G is a finite set 0] points. Then the signature 0] M is given in terms 0] the Betti numbers 0] M by the formula

a(M) = L b4j(M) - L b4j+2(M). j?O j?O

1.6. Cohomology Symplectic and Hamiltonian Torus Actions 35

Proof Let F1, ... ,Fs be the components of MG. Since each Fi is an isolated point, in the notation of Theorem 1.5.9, ri = O.

Let Fi = {xd for 1 ~ i ~ s. By Theorem 1.5.9 with z = A,

s

Lb4j(M) - Lb4j+2(M) = L(-lt-ki .

i=l

On the other hand, by the Atiyah-Singer g-signature Theorem (see, s

e.g., [48], Sec. 5.8), a(M) = LEi, where Ei = ±1, and Ei = +1 if and i=l

only if the orientation on TXi (M) which comes from the circle action is the orientation of M. Thus it remains to show that Ei = (_1)n-k i • This again is a fairly easy consequence of the analysis in [30]. 0

1.6 Cohomology Symplectic and Hamiltonian Torus Actions

Some very interesting results concerning symplectic or Hamiltonian torus actions can be proven very easily directly from the Localization Theorem and cohomology theory: other results, whieh look eohomologieal, require more geometrie reasoning, often Morse-Bott theory. In this section and the next we shall give some results of eaeh type.

The following definition abstracts certain eohomologieal properties of symplectic manifolds.

Definition 1.6.1. Let k be a field of characteristic zero (typicalty Q or R), and let X be a Poincani duality space over k with formal dimension 2n.

(1) X is said to be a c-symplectic (cohomologicalty symplectic) if there is w E H 2(X; k) such that wn i- O. (w is calted the c-symplectic class.)

(2) 1f X is c-symplectic, for 0 ~ j ~ n, consider the cup product homomorphism

i.e., a f----t wja, for alt a E Hn-j(X; k).


X is said to satisfy the weak Lef5chetz condition if

is an isom01'[Jhism. A nd X is said to satisfy the strang or hard Lefschetz condition if w j is an isomorphism for all j. In the latter case X is also said to be c-Kähler (cohomologically Kähler): i.e., X is c-Kähler if X is c-symplectic with the strong Lefschetz condition satisfied.

Of course, closed symplectic manifolds are c-symplectic (over R), and closed Kähler manifolds are c-Kähler (over R). Non-singular complex projective algebraic varieties are c-Kähler over Q. There exist closed symplectic manifolds which satisfy the weak Lefschetz condition but not the strong Lefschetz condition; and there are closed symplectic manifolds which do not satisfy the weak Lefschetz condition: see, e.g., [79].

Let X be a c-symplectic space with w E H 2(X; k) as in Definitions 1.6.1(1). Let G be a compact connected Lie group. Then g*(w) = w, for all 9 E G. Thus any action of a compact connected Lie group on a c-symplectic space is considered to be a cohomologically symplectic action.

Definition 1.6.2. Let X be a c-symplectic space with c-symplectic class w E H 2(X; k). Let G be a compact connected Lie graup, and suppose that G is acting on X. Then the action is said to be cohomologically Hamiltonian (c-Hamiltonian) if

w E im [i* : Hb(X; k) ----+ H 2 (X; k)] ,

where i : X ----+ Xc is the inclusion of the fibre in the bundle Xc ----+ EG.

In view of Proposition 1.5.6, and Remarks 1.5.7(1), an actual symplectic action is Hamiltonian if and only if it is c-Hamiltonian.

As an easy application of the cohomological method we have the following.

Proposition 1.6.3. Let X be a compact c-symplectic space with csymplectic class w E H 2(X; k). Let fd(X) = 2n. Let G = SI be acting on X.

(1) If the action is c-H amiltonian, then there are at least n + 1 fixed points.


(2) 1f X satisfies the weak Lefschetz condition and if XC -::I 0, then the action is c-Hamiltonian.

Proof (1) Choose w E Hb(X; k) such that i*(w) = w. Since i*(wn ) = w n , which is on the top row of the Leray-Serre spectral sequence for Xc ---+ BG, wn is non-torsional. Hence wj is non-torsional for 0 ::; j ::; n; and so, {wj ; 0 ::; j ::; n} generates a free submodule of Hc(X; k) of rank n + 1. By the Localization Theorem, Hc(XC ; k) must also contain a submodule of rank n + 1. Hence dimk H*(XC ; k) 2': n + 1.

(2) In the Leray-Serre spectral sequence for Xc ---+ BG, let d2(w) = tu, where u E H 1(X; k) and t E H 2 (BG; k) is the generator.

Since XC -::I 0, by Proposition 1.4.6(3), d2 (wn ) = O. But d2 (wn ) = nwn -- 1tu. Hence u = 0 by the weak Lefschetz condition. 0

Remark 1.6.4. (1) By Frankel's Theorem (see Corollary 1.5.10), for a Hamiltonian action of G = SI (or G any torus) on a closed symplectic mallifold M, the Serre spectral sequence for Ale: -, ßG collapses (E2 = Eoo ) : i.e., M is TNHZ in MG ---+ BG in the notation of section 2 (discussion following Remarks 1.2.3). This does not hold in general for c-Hamiltollian actions, even in the presence of the weak Lefschetz condition. See [3], Example 1.

On the other hand, it is an old result of Blanchard that given any action of G = SI on a c-Kähler space X, if XC -::I 0, then X is TNHZ in Xc ---+ BG. See [11J or [13J, Chapter XII, Section 6.

Blanchard's result was an important starting point for the cohomology theory of symplectic and Hamiltonian torus actions. Thus he deserves credit for Proposition 1.6.3(2), although the first formally published proof appeared much later [71J.

(2) In contrast to the truly symplectic case, if G = SI acts on a c-symplectic space X, if XC -::I 0, and if F is a component of Xc, then it is possible that F is not c-symplectic. (Cf. Remarks 1.5.12.) In [2J we give an example of a circle action on a c-Kähler space such that no component of the fixed point set is c-symplectic.

The next theorem is another simple, though powerful, illustration of the cohomology theory.

Theorem 1.6.5. Let X be a eompaet c-sympleet-ie spaee, and let G be a eompaet eonnected Lie group.

(1) 1f G ean aet almost - freely on X (i.e., with alt isotropy subgroups finite), then G is a torus.

38 1. Localization Theorem and Symplectic Torus Aetions

(2) If G is aeting almost-freely on X, and if X satisfies the weak Lefsehetz eondition (over a field, k, of characteristic 0), then there is an isomorphism of k-algebras

H*(X; k) ~ H*(G; k) 0 H*(XjG; k).

Proof (1) Suppose that G is acting almost-freely on X; and suppose that G is not a torus. Let K ~ G be a closed simple subgroup of positive rank, and let G ~ K be a subcircle. Let w E H 2 (X; k) be the c-symplectic class. Since H 2 (BK; k) = 0 (see Examples 1.2.1(4)), d2 (w) = 0 in the Leray-Serre spectral sequence for XK ---+ BK. Hence (see just above Remarks 1.2.4), d2 (w) = 0 in the Leray-Serre spectral sequence for Xc ---+ BG. So the action of Gis c-Hamiltonian; and thus XC :I 0 by Proposition 1.6.3(1). This contradicts the assumption that G is acting almost-freely.

(2) Suppose G = T r . In the Leray-Serre spectral sequence for Xc ---+ BG the differential d2 : Eg,1 ---+ ~ is a homomorphism 'IjJ: H 1 (X; k) ---+ H 2 (BG; k). We shall begin by showing that 'IjJ is surjective.

Suppose that 'IjJ is not surjective. Then there is a subcircle K ~ G such that im('IjJ) ~ PK (see Definition 1.3.4). Now consider the LeraySerre spectral sequence for X K ---+ BK, and let H*(BK; k) = k[t]. Since im( 'IjJ) ~ P K, d2 : Eg,1 ---+ ~ is now zero.

Let w be the c-symplectic class, and let d2(w) = tu, where u E

H 1(X; k). By Poincare duality and the weak Lefschetz condition, if u :I 0, there is v E H 1 (X; k) such that w n = w n - 1 vu. From the latter equation it follows that d2 (wn ) = O. Eut d2 (wn ) = nwn - 1tu. By the weak Lefschetz condition, d2 (w) = O. So the action is c-Hamiltonian, and, hence X K :I 0. This contradicts the almost-freeness; and so 'IjJ is surjective.

Now consider the Leray-Serre spectral sequence for G ---+ X ---+ Xc, that is, strictly speaking, G ---+ EG x X ---+ Xc : see Section 1.2. Since this bundle is the pull- back of G ---+ EG ---+ BG along p : Xc ---+ BG, and since the homomorphism 'IjJ, ab ove , is surjective, the spectral sequence for G ---+ X ---+ Xc collapses. To see this, let H*( G; k) = /\(SI, ... ,sr), the exterior algebra with deg(sd = 1 for 1 ::; i ::; T; and let tl,"" t r E H 2(BG; k) be the corresponding generators of H*(BG; k). Because of the pull-back, d2 (Si) = p*(ti)' But p*(ti) = 0, since 'IjJ is surjective.


Choose al, ... , ar E Hl(X; k) such that j*(ai) = 8i, for 1 ~ i ~ r, where j : G ---+ X is the inclusion of the fibre. Since 81, ... ,8r generate an exterior algebra, al, ... ,ar generate and exterior algebra also. Thus there is an injective homomorphism a : H*(G; k) ---+ H*(X; k) given by a(si) = ai far 1 ::; i ::; r.

Since the action is almost-free, Hc(X; k) ~ H*(XjG; k) : see Remarks 1.2.3(1). Thus, where 1f : X ---+ XjG is the orbit map, 1f* :

H*(XjG; k) ---+ H*(X; k) is given essentially by the base edge effect in the spectral sequence for X ---+ XG. Since the spectral sequence collapses, the k-algebra homomorphism H*(G; k) Q9 H*(XjG; k) ---+ H*(X; k) determined by a and 1f* (i.e., u Q9 Y r---+ a(u)1f*(Y)) IS an isomorphism. o

Remark 1.6.6. Theorem 1.6.5 first appeared in [1 J, which was unpublished. It was discovered also by Lupton and Oprea, and it was published in [60J.

The following proposition IS a partial generalization of Example 1.3.12.

Proposition 1.6.7. Let X be a compact c-symplectic space with fd(X) = 2n. Let a torus G act on X in a c-Hamiltonian way. Let F l , ... , Fs be the components of X G , and let fd(Fd = 2ni for 1 ::; i ::; s. Then

s

n + 1 ::; 2:) ni + 1). i=l

Proof Let w E H 2 (X; k) be the c-symplectic class. Choose w E

Hb(X; k) such that i*(w) = w. Let <Pi : F i ---+ X be the inclusion, and let <pi(w) = ai + Yi, where ai E H 2 (BG; k) and Yi E H 2(Fi; k).

s Let z = I1(w - adni+1. Then <p*(z) = 0, where <p : X G ---+ X is

i=l the inclusion. So by the Localization Theorem, z is torsional. Sup-pose deg(z) = 2m ::; 2n. Then wn-rnz is also torsional. But wn-rnz projects to wn on the top row of the Leray-Serre spectral sequence for XG ---+ BG. And wn is not torsional. Thus deg(z) ~ 2n + 2. 0

Remark 1.6.8. Proposition 1.6.7 shows again that there are at least n + 1 fixed points. And, if the action is cohomologically effective, so that ni < n for 1 ::; i ::; s, then the proposition shows that there are at least two fixed point components. It is an interesting theorem, however, that


any effective Hamiltonian action of the ~ on a closed symplectic manifold must have at least r + 1 fixed point components. This is usually proved using the ~ ~ Convexity Theorem: see, e.g., [9], Chapter III, Corollary 4.2.3 and its proof. We shall finish this section with a purely cohomological proof of this result. Since ~

Hamiltonian torus actions, unlike truly Hamiltonian torus actions, are not necessarily TNHZ, we need to impose the condition of uniformity, which is defined as follows.

Definition 1.6.9. Let G be a torus, and suppose that G is acting on aspace X with X G i- 0. The action is said to be uniform if , for any subtorus K ~ G, every component of X K contains a component of XG. 1f X is TNHZ in XG ---t BG, then the action is uniform: see !4}, Corollary (3.6.19). So Hamiltonian torus actions are always uniform.

Theorem 1.6.10. Let G = TT, the r-torus, let M be a closed csymplectic manifold, and suppose that G is acting on M in an effective, uniform, c-Hamiltonian way. Then MG has at least r + 1 components.

Proof Let F I , ... , Fs be the components of MG, and suppose that s :'S r. Let w E H 2 (M; k) be the ~ class, and choose w E

Hb(M; k) such that i*(w) = w. Let <Pi : Fi ---t M,l :'S i :'S s, and <P : MG ---t M be the inclusions. Let <pi(w) = ai + Vi, where ai E H 2 (BG; k) and Yi E H 2 (Fi ; k), for 1 :'S i :'S s. Replacing w by w-al, we can assurne that al = 0. Also, let Xi E Fi , and let ?/Ji : {xd ---t M be the inclusion: then ai = ?/Ji (w) for 1 :'S i :'S s.

Since we are assuming that s :'S r, there is a subcircle K ~ G such that ai E P K for 2 :'S i :'S s. N ow let E l , ... , Ern be the components of M K ; and let (}i : Ei ---t M,l :'S i :'S m, and () : M K ---t M be the inclusions. Denote also by w the image of w E H'k(M; k). Let (}i(w) = bi + Zi, where bi E H 2 (BK; k) and Zi E H 2 (Ei; k). Since the action is uniform, for a given i, 1 :'S i :'S m, Xj E Ei for some j, 1 :'S j :'S s. Let PK : BK ---t BG be the inclusion. So bi = PK?/Jj(w) = PK(aj) = 0, since aj E P K.

Let dim(Ed = 2Ci,1 :'S i ~ m, and let c = max{ Ci, 1 ~ i :'S m}. Since zf+ 1 = 0, for 1 :'S i :'S m, ()* (w c+ l) = O. So wc+ 1 is torsional in H'K(M; k). Hence c ~ n. This contradicts the effectiveness of the action. o

1.7. An Example 41

Remark 1.6.11. (1) In Theorem 1.6.10 we could let M be any compact c-symplectic space, but then we would need to require that the action be cohomologically effective rat her than effective: see [4], Definition (4.7.3) and Remark (4.7.4); see also Theorem 1.4.2 above.

(2) All the results of this section are valid for paracompact finitistic c-symplectic spaces if one adds the condition that the number of connective orbit types be finite. (See Remarks 1.3.8(3).)

1. 7 An Example

In this short section we shall state three interesting theorems which assert that certain symplectic actions are necessarily Hamiltonian. We shall then give a simple example which shows that these theorems are not purely cohomological : their proofs require more geometrical reasoning.

The first theorem is due to McDuff.

Theorem 1. 7.1. ([67}) Any symplectic circle action on a closed fourdimensional symplectic manifold with non-empty fixed point set is Hamiltonian.

The second theorem is due to the Tolman and Weitsman.

Theorem 1.7.2. ([78}) Any semi-free symplectic circle action on a closed symplectic manifold with non-empty finite fixed point set is Hamiltonian.

The third theorem is an easy consequence of the analysis in [67]. (See, for example, [33].)

Theorem 1.7.3. Any effective symplectic action of the n-torus, T n , on a closed symplectic 2n-dimensional manifold with non-empty fixed point set is Hamiltonian.

In her paper, McDuff gives an example of a non-Hamiltonian symplectic circle action on a closed six-dimensional symplectic manifold with non-empty fixed point set. In their paper, Tolman and Weitsman give a substantial amount of information concerning semi-free symplectic circle actions with only isolated fixed points: cohomologically these actions look like the standard diagonal circle actions on prod ucts of two-dimensional spheres. (See also [41].) This suggests the following example.


Example 1.7.4. Let G = K x L where K = L = SI, let X = S2 X S2, and let G act on X by the product action, the action on each factor being the standard rotation. Thus the only isotropy subgroups are G, K, L alld the trivial subgroup. There are four fixed points, (n, n), (n, s), (s, n) and (s, s), where n and s are the north and south poles of each sphere.

Remove small invariant open discs (slices) around (n, n) and (n, s). Now one can attach S3 x I equivariantly to make a closed orientable four-dimensional manifold, M, with a torus action having just two fixed points, (s, n) and (s, s). A straightforward Mayer-Vietoris sequence argument shows that

'( ) {Z for i = 0,1,3,4 H'M;Z = Z EB Z for i = 2

Since H 2 (M; Q) i- 0, M is c-symplectic over any field of characteristic 0.

The fixed point set of K, M K, consists of a torus and a sphere. The toric component of M K does not contain a fixed point of G. Thus the action of Gis not uniform (see Definition 1.6.9). Let H = {(g, g) E G; gE SI}, the diagonal subcircle of G. Then M H = MG = {(s, n), (s, s)}.

The action of neither H nor G is c-Hamiltonian with respect to any c-symplectic class in H 2 (M; Q) by Proposition 1.6.7. And, clearly, the action of H is semi-free.

See [3] and [42] for other examples which show the geometric nature ofMcDuff's theorem. And see [57] for an improvement ofTheorem 1.7.3 in w hich T n is replaced by T n - 1 .

Chapter 2

JLoric "arieties

2.1 Introduction

Anormal variety X which contains as a dense open subset an algebraic torus T ~ (C*) n such that the T action on itself lifts to an action T x X ----+X is called a toric variety. (All varieties we consider are reduced, irreducible and of finite type over C.) Examples of torie varieties are the torus (c*)n, the affine space Cn and the complex projective spaee pn.

Basic to the study of torie varieties is the not ion of fans and their morphisms. Fans are made up of strongly convex rational polyhedral cones. Affine torie varieties eorrespond to rational polyhedral eones and a general torie variety is made up of affine ones, the patching data being given by the eombinatorics of the fan.

Torie varieties were introduced by M. Demazure[26] in connection with the study of the Cremona group and by G.Kempf, F.Knudsen, D.Mumford and B.Saint-Donat [68] in their study of compactifications of locally symmetrie varieties. The subject has sinee then been studied extensively and also has been generalized to obtain new c1asses of manifolds. We refer the reader to the papers of M.Davis and T.Januszkiewicz [25] and M.Masuda [62]. See also [18] and [23] and the survey articles [22] and [70].

In these notes, we give a quick introduction to the subject. There are several sources available whieh give eomprehensive treatment of the subject such as the survey artic1e [24] and the books [69], [29] and [32]. We shall follow the book [32] c1osely.

44 2. Toric Varieties

2.2 Affine torie varieties

We start with the definition of a rational polyhedral cone. Let N be a free abelian group of rank n, and let M =

Homz(N, Z). Denote by N R the real vector space N ®z Rand by M R the dual vector space M ®z R.

One has the canonical pairing (.,.) : M x N ~ which extends to a pairing (.,.) : MR x N ~ Having chosen a Z- basis el,· .. , en

for N , we denote the dual basis for M by ei, ... ~ When N = zn C

Rn = N R, el , · . . ,en will denote the standard basis of N. A polyhedral cone is a set

where VI , ... , Vt E N R. We say that a is a rational polyhedral cone if all the Vi can be chosen to be in N. The Vi are called the generators of the cone. The lattice N is also part of the data for a rational polyhedral cone. For this reason, we need to distinguish between a rational polyhedral co ne a in N and the underlying point-set in N R that it determines. The latter will be denoted lai.

For a cone adefine the dual cone to be the set

a V = {u E MR I (u , v) 2: 0 V 11 E a}.

Then a V is a cone in MR . Furthermore, a V is a rational polyhedral cone in M if a is a rational poly hedral cone in N.

We say that a is a strongly convex rational polyhedral ("scrap" ) cone if a is a rational polyhedral co ne such that an (-a) = o.

Let a be a rational polyhedral co ne in N and let Su eMdenote the set

Su:= a V nM.

This is a subsemigroup of M which is finitely generated as a semigroup. We prove this at the end of this section. Thus, the semigroup algebra Au := C[SuJ is a finitely generated algebra over C.

For a strongly convex rational polyhedral cone a in N, we define Uu to be the affine variety Spec(Au). Since Su e M, Au is a subring of Ao = Z[MJ. Hence Au is an integral domain. This implies that Uu is an irreducible affine algebraic set. We shall show below that Uu

contains T = Hom(M, C*) ~ (c*)n as a dense open set and that the multiplication of T lifts to an action of T on Uu . We shall prove in

2.2. Affine toric varieties 45

seetion 2.5 that Ua is indeed anormal variety. An irreducible normal affine variety U which contains T as a dense subvariety and on which T acts in such a way that the T action restricts to the multiplication on T c U is called an affine toric variety. Thus Ua is an example of an affine toric variety. It is true that any affine toric variety U arises as Ua for some strongly convex rational polyhedral cone a. See Remark 2.5.11.

A general toric variety is obtained by patching together a collection ~ where b. is a fan in N. See §2.3 for the precise definition.

Example 2.2.1. (i) Let a = {O} be the O-dimensional co ne in N. Thus a V = MR. Hence Aa = Ao is the group ring C[M] of M which is the Laurent polynomial ring C[Xi ,Xi- l I1 ~ i ~ n] where Xi = Xe:. The algebra Ao is obtained from the polynomial algebra C[X 1, ... , X n] by localizing with respect to the multiplicatively closed set {uklk ~ O} where u = Xl' .. X n. It follows that Uo = Spec(Ao) is obtained from Cn = Spec(C[X1 ,··· ,Xn]) by removing the zero locus ofu. Thus

(ii) Take a = R>Oel + R>Oe2. Then a V = R>oei + R>oe2' Hence Sa = Z2':oei + ~ ~ Aa = C[Xej , Xe;], and so Ua = A2, the affine plane.

A face T of a rational polyhedral cone a in N is a cone in N such that ITI = lai n ul. for sOme u E a V • Any cone a is a face of itself. We write T ~ a to denote that T is a face of a.

If ais ascrap cone, then so is any face of a . Note that if T ~ a, then T V ~ a V . Hence ST ~ Sa. It follows that C[ST] ~ C[Sa]. This leads to a canonical morphism Spec(C[STD = UT ~ Ua = Spec(C[Sa]). In fact the above morphism expresses UT as a principal open subset of Ua· Indeed, one can show that ST = Sa + Z>o( -u), where u E M is chosen to be in the relative interior of a v n Tl.. (See Prop. 1.2, [32].) Hence AT = C[ST] has aC-basis consisting of elements of the form Xa- ku = xaj(xu)k,a E Sa,k E Z2':o. Thus AT = (Aahu, and UT = {x E Ualu(x) i=- O}. In particular note that, for any scrap cone a, Ua contains Uo = T as a dense open subset.

A point of Ua (we consider only closed points) corresponds to a maximal ideal in Aa which in turn corresponds to the kernel of an algebra homomorphism from Aa to C. Since our algebra actually arises in a


special way, viz, as a semigroup algebra over C, such algebra homomorphisms correspond in a natural way to semigroup homomorphism Ser --7 C* U {O} = C, where the semigroup structure on Cis induced by multiplication. The semigroup homomorphisms are required to preserve units: Ser ::1 0 f--t 1 E C. We remark that if u E Ser is mapped to 0 under such a semigroup homomorphism then u can have no inverse in Su.

Let Ser be generated as a semigroup by UI, ... ,Uk. Then Aer is generated as an algebra by Yi = XUi , 1 ::; i ::; k. Hence A er is expressible as a quotient C[Yi,'" ,Yk]1 I of the polynomial algebra C[YI,'" ,Yk]' Geometrically, this corresponds to imbedding Uu in the affine space Ck 2:: Spec( C[YI , ... ,YkJ) as the zero set of the functions in the ideal I. In this way a point (al,'" ,ak) E Ck is in Uu if and only if the map Uj f--t aj E C extends to a morphism of semigroups Ser--7C preserving units.

We shall lift the multiplication action of T on itself to the whole of Uer . First note that the multiplication map T x T--7T induces the map AO--7Ao @ Ao defined by Xi f--t Xi @ Xi. If t E T, x E T are viewed as semigroup homomorphisms, then their product tx is the semigroup homomorphism M--7C defined as tx(u) = t(u)x(u). We define for tE T, and x EUer, the product tx to be the semigroup homomorphism tx(u) = t(u)x(u) E C. This defines an action of Ton Uer . This corresponds to the algebra map Ao- --7 Ao @ Au defined by XU f--t XU @ XU for u E Ser , where we naturally regard Ao- as a subalgebra of Ao. Thus Uer is an affine toric variety. If T is a face of a , then T -action on U er restricts to the natural T-action on UT •

Example 2.2.2. Let a be the scrap cone R?Oe2 + R::::o(3el - e2). Then So- is the semigroup generated by ei , ei + 3e2' ei + 2e2' ei + e2' Aer is then the algebra C[X, XY, Xy 2, Xy 3] c C[X, Y], where we have written X = Xej , Y = Xe;. We see that Au = C[X, U, V, WJ/ I, where the ideal is generated by XV -U2, XW -UV. Thus the variety Uer can be identified as the subvariety {(x, u, v, w)lxv = u2, xw = uv} C C4 . The T action on Uer can be described in terms of this embedding as follows : Let (tl, t2) E T, p = (x, u, V, w) EUer' Then t.p = (tIX, tlt2u, ~ ~ E

Uo-.

We shall now establish the following :

Lemma 2.2.3. (Gordan's Lemma) Let a be a rational polyhedral cone. Then Ser = a V nM is jinitely genera ted as a semigroup.

2.3. Fans and Toric Varieties 47

Proof: Let Ul,'" , Uk E M generate a V as a cone in M. Let K = U=l <i<k aiui j 0 ::; ai ::; 1} C a v. Then K is compact. It follows that KnfVfls finite. We claim that KnM generates Ser. To see this first note that Ui E K n M for all i ::; k. Write any U E Ser as U = 2:l<i<k SiUi =

2:[Si]Ui + u' where u' E K. But u' = U - 2:[Si]Ui E M, that Ts; u' E M. Hence u' E K n M. Thus we have expressed u as a non negative integral linear combination of the finite set K n M. 0

Remark 2.2.4. Let U = SpecA be an affine variety (over C) where A = C[Yl , . .. , Yk]1 I is reduced. One obtains an imbedding U C Ck where a point p = (Yl,'" , Yk) ECk is in U if and only if f(Yl,'" , Yk) = 0 for all f E I. Besides the Zariski topology on U, we obtain the analytic topology on U whieh is just the topology on U indueed by the standard Euelidean topology on C k . This topology is also called Hausdorff topology. Thus one has also the analytic topology on Uer for any serap eone a in N.

2.3 Fans and Torie Varieties

We shall establish the eorrespondenee between fans and torie varieties. The morphisms between fans induee equivariant morphisms between eorresponding torie varieties.

Definition 2.3.1. A fan .6. in N is a nonempty collection of scrap cones such that (i) a E .6., and T < a implies T E .6., (ii) a, a' E .6. implies an a' is a face of both a and a'.

Let .6. and .6.' be fans in N and N' respectively. A morphism 'P : .6. ---t .6.' is a homomorphism of lattices 'P : N ---t N' such that for each CJ E .6., there exists a a' E .6.' such that 'P(jaj) C ja'j.

Note that a fan (in N) is determined by listing all the maximal dimensional eones (eaeh of whieh should be ascrap eone) eontained in it. Let j.6.j = UerEb.jaj C NR.

Example 2.3.2. (i) Let a be any serap cone in N. Let .6.er denote the eolleetion of all faees of a (including a itself). Then .6.a is a fan in N.

(ii) Let .6. be the fan in N = Z2 C R2 determined by eones ai =

R?oei + R?O(el + e2), i = 1,2. Then.6. = {al,a2,Tl,T2,T3,O}, where Ti = R?oei, i = 1,2 and T3 = R?o(el + e2).


Remark: We emphasise that the lattice N should also be specified to complete the data for a fan. Similarly a morphism of fans is not determined merely by knowing the map between the cones constituting the fans. For example: Let N = Z, ~ = ~ = {O}. The morphism <p : ~ ~ defined by the map N ----t N, n H 2n is not the same as identity morphism of the fan ~ which is defined by the identity map of N. Indeed the variety associated to both ~ and ~ is C* and the map <p induces the doublecovering map Z H z2 whereas the identity map of the fans induces the identity map of C*. One has a category of fans and their morphisms. We associate to each fan a toric variety and to each morphism of fans a morphism of toric varieties. We shall first define the toric variety associated to a given fan. Let ~ be a fan in N. For each (J E ~ one has an affine variety Uu := Spec(Au). We patch together the various Uu, (J E ~ using the combinatorics of the fan ~ Indeed, if T is a face of (J, then we have seen that UT is an open subvariety of Uu . At the level of co ordinate rings this corresponds to the inclusion Au C AT.

The reader may recall from §2, eh. v, [74], the notion of a scheme obtained by "pasting of schemes" .

Definition 2.3.3. We define ~ to be the scheme whose underlying topological space is UUEß U u where we identify UT, T = (J n (J', as the principal open subvariety in Uu as welt as UU'. The structure sheaf Ox is defined by the requirement that Ox!Uu = Au for each (J E ~

The scheme ~ is irreducible since T = Uo is irreducible and dense in X ~ Furthermore X ~ is reduced, as each Au is an integral domain and hence has no nilpotents. Note that since the natural T-action on Uu restricts to that on UT for any T < (J E ~ one obtains a well-defined (algebraic) T-action on the whole of ~

The variety ~ is separated (which is the analogous notion in algebraic geometry to that of Hausdorffness in topology). This would follow if we show that the canonical map Uu n UU'----tUU X UU' is closed for any (J, (J' ~ Equivalently, one shows that the canonical map Au (9 Au' ----t Aunu' is surjective using the fact that Su + Su' = Sunu' ·

Just as in the case of the affine toric varieties, one has also the analytic topology on ~ induced from the analytic topology on each Uu and the union topology on ~ = UUEß UU • Thus a set W C ~ is open in the analytic topology if and only if Uu n W is open in the analytic topology of Uu for each (J E ~ When ~ is nonsingular as a variety, it is a complex analytic manifold under the analytic topology.


Example 2.3.4. There are only four fans in N = Z. Theyare

The corresponding toric varieties are

C*, C = Spec(C[X]), C = Spec(C[X- I ])

and

pI = Spec(C[X]) USpec(C[X,X-l]) Spec(C[X-- I ]),

the complex projective space of dimension 1 respectively.

Example 2.3.5. Let ß be the fan in N = Z2 C R2 determined by the scrap cones a = R?oel + R?oe2, a' = R?o( -et} + R?oe2. Let T = an a' = R>Oe2. Then S(T = Z>oe; + Z>oei, therefore A(T = C[XI ,X2], U(T = ~ ~ C2, wher; Xj = xe;,j = 1,2. Similarly, S(TI =

Z>oe; + Z>O( -ei)' and so A(TI = C[X11 , X 2] ~ C[X1, X 2]. Therefore ~ ~ C2.- Also Sr = Z>Oel + Z>o(-et} + Z>Oe2 , and hence Ar =

C[XI , xlI, X2]. Therefore-Ur = c*-x C . The ~ X(ß) is obtained by taking c2 U c2 and identifying a point (Xl, X2) E Ur with the point in U(T with coordinates (Xl, X2) and the point in U(TI with coordinates (xII, X2)' The resulting space is the same as the product pI x C .

Note that if ß and ß' are fans in N and N' respectively, then the collection ~ x ~ = {a x a' I a E ~ a' E ~ forms a fan in N x N' .

Lemma 2.3.6. Let ~ and ß' be fans in N and N'. For the product fan ~ ~ in N x N', one has an isomorphism X(ß x ß') ~ X(ß) x ~

Proof: A cone in ß x ~ is of the form T = a x a', with a E ~ E ~

The variety Ur = Spec(Ar ) = Spec(C[Sr]) where Sr is the semigroup T V nA1 x M'. Note that T V = MR X aN nav x Mk = a V x aN. Therefore Sr = S(T X S(TI. It follows that Ar = C[S(T x S(T/] = C[S(T] ® ~ ~ A(T ® A(TI. Thus Ur = U (T x U (TI. These isomorphisms patch together to yield X(ß x ß') ~ X(ß) x X(ß'). 0

Example 2.3.7. Let a be the scrap cone in N generated by a basis VI, ... , V n for the lattice N. Then, writing ~ for the fan in N consisting of all faces of a, one has X (ß(T) ~ cn.


Let ~ be a fan in N. For each IJ E ~ we obtain an action of Ton U(}". Suppose that T is a face of IJ. The T action on UT is the same as that obtained by restriction of the T action on U(}". Thus we obtain a consistent action of T on X ~ In fact we have an action morphism f-L : T x ~ ----t ~

Let <p : N ----t N' be a homomorphism of abelian groups and let <p v : M' ----t M denote the transpose of <po One has TN = H omsg(M, C) = Homz(M, C*). (Here the suffix "sg" stands for "semigroup.")Therefore <p induces a morphism ip* : TN ----t TN' defined as TN :3 x f---t x 0 ipv E

TN,· Let ip : N ----t N' define a morphism of fans ~ ----t ~ We have

a homomorphism of tori ip* : Uo = TN ----t TN' = ~ We shall now construct a morphism of toric varieties ~ ----t ~ which extends ip* on the open set Uo C ~ and intertwines the TN action on ~

and the TN, action on ~ via the homomorphism ip* : TN ----t TN'. For IJ E ~ choose any IJ' E ~ such that </J(IIJI) C IIJ'I. Then ipv

maps ~ C M' into S(}" C M. For x E U(}" = Homsg(S(}",C) define ip*(x) to be the semigroup homomorphism x 0 ip v. This defines a morphism of varieties ip* : U(}"----tU(}"I. Suppose that T is a face of IJ, and IJ' is a face of T' E ~ so that U T C U(}", and U(}"I C UT,. It is trivial to check that ip* : UT----tUT, is same as the restriction of the morphism U(}"----tU(}"I.

Therefore we obtain a weIl defined morphism ip* : ~ ----t ~ Let t E TN, xE U(}" and u' E S(}"I. Then

ip*(t.x)(u') = ((t.x) 0 ipV)(u') = (t.x)(ipV(u'))

= t(ipv(u')).x(ipV(u')) = ip*(t)(u').ip*(x)(u').

This shows that the morphism ip* intertwines the action of TN on X ~

and that of TN' on X ~ via the induced map ip* : TN ----t TN,.

Example 2.3.8. Let N = zn and let lJi denote the co ne generated by the set {eI,··· ,en , eo = -(eI + ... +en )} \ {eil, 0::; i ::; n. Let ~ be the fan in N consisting of all the lJi and their faces. Then, writing Ui for U(}"i' we have Ui = Spec(Ai ) ~ cn where Ao = Aao = C[XI,··· ,Xn] and Aj = C[Xj-l,xIXj-l, ... ,XnX;l],j i= O. For 0::; i < j::; n, A(}"in(}"j =

AdXiX;l] = Aj[Xi- 1 Xj]. The toric variety ~ is just the projective space pn where an element x E pn with homogeneous coordinates (zo : Zl : ... : zn) is identified with the n-tuple (ZO/Zi,··· ,zn/zd E Ui ~ cn if Zi i= 0,0::; i ::; n. For t = (tl,··· ,tn) E TN = (c*)n, x E ~ = pn, the element t.x has homogeneous coordinates (zo : tlZl : ... : tnzn).


Example 2.3.9. Let CY be a scrap cone in N. Let N cr be the subgroup generated by CY n N, and let N(cy) denote the quotient N/Ncr . Denote by ~ the fan in N consisting of all faces of CY. Let ~ denote the fan {O} in N (CY). We have an obvious morphism from ~ to ~ defined by the canonical quotient map rJ : N ---+ N(cy). The induced morphism Ucr = ~ ---+ ~ = TN(cr) is the projection of a fibre bundle with fibre the toric variety corresponding to the scrap cone cy' which is just Icyl thought of as a co ne in N cr . Indeed choosing a splitting rjJ : N(cy) ---+ N for rJ, we obtain an isomorphism N ~ N cr EB N(cy), v H (v - rjJo rJ(v), rJ(v)). This is a morphism of fans ~ ~ ~ X ~ and so we obtain ~ = Ucr ~ Ucr' x ~ = Ucr' X TN(cr)' The morphism X ~ ---+ X ~ then coincides with the second projection map.

j ~ p Let 0---+ N' y N --'-+ N ---+0 be an exact sequence of lattices. Let

~ ~ l be fans in N, N', N respectively such that the morphisms of lattices are in fact morphisms of the respective fans. Thus we have

induced morphisms ~ Ä X(l) ~ ~

Proposition 2.3.10. Suppose that ~ C land that there exists a fan ~ C l such that the following conditions hold: (i) for each cone Ci E l there exist unique cones cy' E ~ and f E ~ such that Ci = cy' + f,

(ii) if cy' E N and f E ~ then Ci := cy' + f E l, and, (iii) ~ is a lift ~ i.e., each cone T E ~ is the image of a unique cone f ~

Then p* : ~ is the projection of a Zariski locally trivial fibre b1lndle with fibre ~

Proof: ~ T E ~ and let f E ~ be its "lift." Choose a splitting rjJ : N ---+N of p such that Ifl c rjJ(N)R. Thus ~ is a fan in rjJ(N) (cf. 2.3.2). The collection of all cones in l spanned by cones in ~ and faces of f is a fan in N, ~ by ~ The fan ~ consists precisely of those cones in ~ with image contained in ITI. The map p : N ---+ N is a morphism of fans p -1 ~ ) ---+ ~ The ind uced morphism of toric varieties is just the restrietion of P* to p-;l(UT ).

On the other hand the isomorphism J; : N ~ N' EB N, defined as V H (v - rjJ 0 p(v),p(v)) is a morphism of fans ~ ~ X ~ We have a commuting diagram

52

1 id

-----+

2. Toric Varieties

1 This shows that p* : X (3.) -----+ X (ß) is a locally trivial bundle such

that {UT ~ is a trivializing open cover. 0

Example 2.3.11. Let n 2 2 and let N = zn,N' = Zv, where v =

el + ... + en and N = N IN'. Let ß' = ~ O}, 3. the fan consisting ofthe cones ai generated by V,el,'" ,ei-l,ei+l,'" ,en , 1:::; i:::; n and their faces. Let ß ~ the fan in N consisting of those cones whk!I are images of cones in ß. For 1 :::; i :::; 71, let Ti be the co ne in N generated by ej, 1 :::; j :::; 71, j i- i. Taking ~ to be the fan consisting of all the Ti and their faces, the hypotheses of Proposition 2.3.10 are easily verified. We know that X(ß') = C. The variety X(ß) is the projective space pn-l as can be verified directly. X(3.) is the blow-up of cn at the origin (see Example 2.5.6). The resulting bundle X(3.)-----+pn-l is just the tautological bundle O( -1).

2.4 Polytopes

By a (convex) polytope P we mean a subset of areal vector space E which is a convex hull of a finite number of elements of E - called vertices of P. A supporting hyperplane for P consists of all v E P determined by a vector u E E* and r ERsuch that (u, v) 2 r. A face of P is the intersection of P with a supporting hyperplane. We regard P as an improper face of itself. We shall assurne that dirn P = dirn E, and that 0 is in the interior of P. Let N be a lattice in E. We say that p is a rational polytope if the vertices of P lie in N.

The polar of Pis the set po = {u E E*I(u,v) 2 -IYv E P} c E*. For any polytope P, po is a polytope in E* and p oo = P. Furthermore, if Fis a face of P, then F* = {u E POI(u,v) 2 -l,Vv E F} is a face of po. The association F f--7 F* establishes an order reversing bijective correspondence between faces of P and those of F*. If P is a rational polytope with respect to a lattke N in E, then po is rational with respect to the duallattice M = Homz(N, Z) C MR = E*.

Starting with a rational polytope P, we obtain a fan ß by taking cones over the proper faces of P. Since the origin is in the interior of

2.5. Smoothness and Orbit Structure 53

P, the resulting fan is complete. More generally one could carry out the above construction by taking cones over the proper faces of a sub division of P.

Now let N be a lattice in areal vector space V of dimension n, and let M denote the dual lattice in V*. Let P be any rational polytope of dimension n. We do not assurne that P contains the origin in its interior. For any nonempty face Q of P denote by (JQ the cone {v E

VI(u,v) :::; (u',v), V u E Q, V u' E P}. Note that (JQ is a scrap cone; this is because P is rational and is of the same dimension as V*.

Lemma 2.4.1. The collection b..p = {(JQ I Q is a non-empty face of P}, forms a fan. If P contains the origin in its interior, then b.. p is the fan consisting of cones over the proper faces of po. 0

2.5 Smoothness and Orbit Structure

Let b.. be a fan in N. We wish to give a combinatorial criterion for smoothness of the toric variety X (b..). Since smoothness is a local property, it suffices to consider the affine toric variety Uu for a scrap cone (J.

Let X u : Su--+C denote the semigroup homomorphism defined as xu(u) = 1 if -u E Su, and xu(u) = 0 if -u rt. Su. Note that ifu, u', -(u+ u') E Su, then -u, -u' E Su, and so X u is a well-defined homomorphism of semigroups. Thus X u E Uu .

We claim that X u is a (T = )TN-fixed point if (J is a cone of dimension n = rank(N). To see this note that since (J is maximal dimensional, u E

Su implies -u rt. Su unless u = O. For t E TN , txu(u) = t(u)xu(u) = 0 for all u E Su, u -I- O. Of course, tu(O) = 1 = t(O)xu(O). Hence X u is T-fixed. Suppose x E Uu is T-fixed. We must show that x(u) = 0 for all nonzero u E Su. Suppose, on the contrary, x(u) = z -I- 0 for some u E Su. We may take u to be a primitive vector in M. Now this u can be extended to a basis for M. Hence there exists a t E T such that t( u) = 2. For this t, clearly tx( u) = 2z -I- x( u). This contradicts the assumption that x is T fixed. Hence X u is the unique T-fixed point.

Lemma 2.5.1. Let (J be a scrap cone in N. Uu is smooth if and only if (J is generated as a cone by part of a basis for the lattice N.

Proof: Suppose (J is a scrap cone in N, dim((J) < n = rank(N). Then Uu

can be fibred over a torus by an affine toric variety which corresponds to


the cone 10-1 with lattice structure Nu. (See Example 2.3.9.) Therefore we may assume that 0- is n-dimensional.

Now if 0- is spanned by a basis for N, then by Example 2.3.7 we know that U u is isomorphie to the affine space Cn . Hence U u is smooth.

Now assume that 0- is not generated by part of any basis for N. We shall show that X u is not a smooth point in X = Uu .

Indeed, let m denote the maximal ideal in the local ring 0 x ,Xo-. Recall that C[X] = C[XU : U E Su] c C[M]. For any U E Su,u -=I 0, XU E OX,xo

is in the maximal ideal m because XU(xu ) = xu(u) = 0. Also if u E Su

can be expressed as UI + U2,UI,U2 E Su, then XU1 +U2 = XU1 XU2 E m2 .

Furthermore if U E Su is the first lattice vector in R?ou C 0-v, then XU cannot be expressed as a product of two elements of m, i.e., XU ~

m2 . (This is because Au is an Su-graded algebra.) This implies that dimension of m/m2 is at least as big as the number of edges of the co ne o-v. In particular X u is smooth only if o-V is generated by n edges. (Since 0- is maximal dimensional, 0-v will be generated by at least n edges.) The dimension of m/m2 is n only if the first lattiee points along the edges of o-V generate Su as a semigroup. This implies that these lattice points form a basis for 0-v. Equivalently, X u is smooth only if 0- is generated by a basis for the lattice N. 0

Corollary 2.5.2. Let ~ be any fan in N. Then ~ is smooth if and only if each 0- E ~ is generated as a cone by part of a basis for N. 0

Definition 2.5.3. A scrap cone in N is called nonsingular if it is generated by part of a basis for N. A scrap cone of dimension k is called simplicial if it is generated by exactly k edges. A fan is called nonsingular (resp. simplicial) if every cone in it is nonsingular (resp. simplicial).

Every 2- dimensional cone is simplicial. However this is not so for higher dimensional cones. (For example consider the co ne over a square in R3 .)

Suppose that 0- is a scrap cone in N which is simplicial. Then it is easy to see that there exists a sublattice N' C N such that 0-

is nonsingular with respect to N'. Indeed, suppose that VI,··· ,Vk E

0- n N are the first lattice points along the edges of 0-. Let Nu denote the subgroup of N generated by the Vi. Then Nu can be extended to a lattice N' C N such that 0- n Nk = Nu. Clearly, with respect to N', 0- is regular. Let us denote by 0-' the scrap cone 10-1 with the N' lattice structure. One has a morphism of affine toric varieties


X' := ~ =: X induced by the inclusion N' ~ N. Denote by G the group N/N' = Homz(M'/M,Q/Z} = Homs.g(M'/M,C} c Hom(M', C*}. This group can be idelltified with the kernel of the homomorphism TN, = Hom{M', C*} ~ TN = Hom{M, C*} induced by the inclusion of N' in N. Note that, if 9 E G, then g{u} = l\iu E M. Restrict the action of TN, on X' to the subgroup G. We claim that X is the quotient of X' by the action of G. To establish the claim we have to show that C[X] = C[X']G. Recall that C[X'] = Aa' = C[xUlu E Sa/] ::) C[xUlu E Sa] = Aa = C[X], where Sa = (Tv nM c (Tv n M' = Sa" Therefore we must show that A a = ~ We shall first establish that Aa C ~ Indeed, if 9 E C, u E Sa,x' EX', then (g·XU)(x') = XU(g-lx') = g-i(u}.x'(u} = x'(u} = XU(x'}. Thus Aa C ~ Now let f rf:- A a · Write f = 2: CjXUj for some Cj E C, Uj E M'. Since f rf:- Aa , at least one Uj, say Ui is not in M. Choose ag E G such that g(ud i= 1. Then, it is straight forward to check that g(f} i= f. Therefore, AlT = ~

Suppose that L\ is a fan in N in whieh every cone is simplicial. Then X (L\) is covered by affine open sets U a each of w hieh i:,; t, quotient of affine spaces by a finite group. Thus, locally, X(L\} is a quotient of a smooth variety by the action of a finite group. Such a spaee is called an orbifold. In general, X(L\} may not itself be a quotient of a smooth variety by the c.ction of a finite group.

Example 2.5.4. Let no,' .. ,nk be a sequenee of positive integers such that ged(no,' .. ,nk} = 1. Consider the fan L\ in Zk eonsisting of the k dimensional eones (Ti, 0 ::; i ::; k and their faces. Here (To is generated by the standard basis veetors. For i > 0, (Ti is generated by ej, j i= i, together with v = -(ei + ... + ek). The corresponding torie variety is the projective spaee p k. Now replaee the lattice N = Zk by the lattiee N' spanned by (l/ndei,l ::; i ::; k, (l/no}v. Let L\' be the fan in N' having the "same" eollection of eones as L\ thought of as eones in the lattice N'. Then the resulting torie variety is the weighted projeetive spaee X = P(no,'" ,nk). The group G = N' /N = (J)O<i<kZ/niZi acts on p k by (.[zo : ... : Zk] = [(ozo : ... : (kZk] where ( = ~ .. · ,(k) E G. The weighted projeetive spaee X is just the quotient of pk by the action of G and the morphism pk ~ P{no,'" ,nk) is the morphism of toric varieties induced by the morphism of fans L\ ~ L\' eorresponding to the inclusion Ne N'.

Although an arbitrary toric variety may not be smooth, every torie variety is normal and Cohen-Maeaulay. We shall only establish normality of a torie variety.


Recall that an affine variety X is normal if C[X] is integrally closed in its quotient field. An arbitrary variety is normal if it is covered by affine open varieties each of which is normal.

Let .6. be any fan in N. Since X(.6.) is covered by U,n a E .6., it suffices to show that for any scrap co ne a in N, Ua is normal. Let Tj, 1 :S j :S k denote the edges of a. Then Sa = nl-:::'j-:::'kSTj. It follows that A a = C[Sa] = nl-:::'j9C[Sr;l = nl-:::'j-:::'kATj. Since Tj is a one dimensional cone, clearly, ATj is isomorphie to

which is integrally closed. (Alternatively, any one dimensional cone is nOllsingular and so UT) is smooth. In particular UTj is normal and so ATj is integrally closed.) It follows that Aa is also integrally closed.

We now describe a criterion for compactness of a toric variety in terms of the combinatorics of the fan determining it. A variety X is compact in the analytic topology if and only if it is complete. A variety X is called complete if the morphism X ---+Spec( C) is proper. Properness of a morphism p : X ---+ Y can bc verified using the valuative criterion: Let R be any discrete valuation ring and let i : R c K denote the inclusion of R into its fraction field K. The morphism pis proper if and only if for any morphisms 10 : ~ 1 : Spec(R)---+Y such

~ 010 = 1 ~ i*, there exists a morphism 1 : Spec(R) ---+ X such that po 1 = 1 and f 0 i* = fo·

Spec(K) ~ X

i* 1 /' lp Spec(R) ---+ Y.

f

Properness of p : X ---+ Yensures that the fibres of p are compact and the image of p is closed.

We state without proof the following, referring the reader to §2.4, [32] for details. See also §9, eh. VI, [29].

Theorem 2.5.5. Let <p : N ---+N' be a morphism of fans .6.---+.6.'. The

induced map <p* : X(.6.)---+X(.6.'), is proper if and only if <p- 1(1.6.'1) = 1.6.1. In particular, X(.6.) is complete if and only if UaEfllal = NR. 0


Example 2.5.6. An impürtant example üf a_proper mürphism is that üf a blow up. Let X = Cn. The blüw up p : X -+ X üf X at the ürigin is the restrictiün üf the secünd projectiün map X X pn-l-+ X where X C X X pn-l is the variety defined by the vanishing üf the füllüwing set üfhümügeneüus pülynümials: XiZj-XjZi E C[XI ,··· ,Xn, Zl,··· ,Zn]' 1 ::; i, j ::; n. Here Xi are the usual cüürdinates in cn and Zl, ... ,Zn are the hümügeneüus cüordinates in pn. Let Ui = {(x, z) E Cn X pn-Il zi =1=

O},l ::; i ::; n. Let Yi = Xi and 1j = Zj/Zi,j =1= i. Ui-+cn, Ui :3

(x, Z) t---7 (YI,··· ,Yn) is an isümürphism üf varieties. The Ui cüver X. We let T = (c*)n act .on X and pn-l the usual way: Für t E T, x E X, Z E pn-l, we define t.x = (tlXl, ... ,tnxn) and tz = (tlZI : ... : tnzn). ~ that X is stable under the T actiün_ .on X X pn-l and that the map X -+X is T equivariant. 'Ehe variety X is in fact a türic variety. The reader shüuld verify that X is just the X(ß) cünstructed in Example 2.3.8.

Let ß be a fan in N. We wish tü describe, in terms üf the cümbinatürics üf the fan ß the T = TN .orbits üf X(ß). It is instructive tü cünsider the situatiün in the simplest example, namely, Cn = Ua where a is the nünsingular cüne generated by the standard basis vectürs el, ... ,en in N = zn. Of cüurse the .orbit clüsures in this are the affine spaces Cl, I c {I, ... ,n}, spanned by the basis vectürs ei, i EI. Indeed, denüting by Xi the element Xei , für any face T üf a, xT(Xd = 1 if -ei E ST' and XT(Xi) = ° ütherwise. But -ei E ST if and ünly if ei tt- ITI. The Türbit üf the püint XT is the set OT = {txT It E T}. Thus, für any x E On Xi(x) =1= ° if and ünly if ei tt- T The cürrespünding .orbit clüsure is V(T) = {(Xl,··· ,Xn)IXi = O,ei E ITI}. Thus we have a bijectiün a > T B V(T) = {(Zl,··· ,Zn)IZi = O,ei E ITI}. This bijectiün is .order reversing: T < TJ implies V (T) =:l V (TJ). Alsü, nüte that, each .orbit clüsure VT ~ Cn - k , k = dim( T) is again a toric variety.

Müre generally, let ß denüte an arbitrary fan in N. Let X(ß) denüte the cürrespünding türic variety. Since für any a E ß, Ua C X(ß) is stable under the actiün üf T, and since the Ua , a E ß cüver X(ß), any T-ürbit üf X(ß) is cüntained in Ua für süme a E ß. In particular, UT<a OT C Ua where OT denütes the .orbit üf XT E UT. We claim Ua =

~ OT· Indeed let x E Ua. Let T be the smallest face üf a such that x E-UT. We claim that x E OT. Tü see this we regard x as a semigroup hümümürphism x : Sa -+ C. Since x E UT C Ua, it füllüws that x extends tü a semigroup hümümürphism x : ST -+ C. Since x tt- UlI

für any prüper face TJ < T, we must have x(u) = ° für all u E ST such


that -11 r/:- Sr. (Otherwise, X would extend to Sr + Z>o( -11) and hence X E Uq where 'I = T n ~ Let 111, ... ,11k be the semigroup generators of Sr, and let Zi = x(l1d, 1 ::; i ::; k. Assume that -l1i E Sr, 1 ::; i ::; l, and that -l1i r/:- Sn i > l. Then Zi i= 0 for 1 ::; i ::; land Zi = 0, for i > l. One checks that there is a well-defined semigroup homomorphism t : M ---+ C* such that t(l1d = Zi, 1 ::; i ::; l, and t(l1i) = 1,l < i ::; k, so that t E T. Clearly, tXr = x.

Denote by V(T) C ~ the closure of the orbit Or. Let T < (T. It turns out that V(T) is a T(T) toric variety, where T(T) denotes the torus corresponding to the lattice N(T) = N/Nr . Here Nr is the lattice generated by IT I n N. The fan in N (T) that corresponds to V (T) is the star of T in ~ written as start:;. (T) (or simply, star( T)). By definition, star(T) is the collection of those cones ij in N(T) which are images of (T E ~ under the quotient map N R ---+ N ( T) R such that (T =:J T. (Check that such a ij is indeed strongly convex.) Indeed, we define V ( T) to be the toric variety corresponding to the fan star ( T) in N (T) and prove that V(T) is the closure of the orbit Or in ~ However, this imbedding is not induced by a morphism of fans ~ in general. Note that dim(V(T)) = n - k, where k = dim(T), n = rank(N).

The quotient map N ---+ N(T) induces an inclusion M(T) = Mn ~ ---+ M. Any semigroup homomorphism X : M(T) ---+ C* extends

to a unique semigroup homomorphism x : Sr ---+ C which maps 11 E Sr to zero if 11 r/:- M ( T). This is a well-defined homomorphism because if 11,11' E Sr, 11 + 11' E M(T), then 11,11' E M(T). Note that this embeds T(T) into Ur as the T-orbit of Xr E U(T). If (T =:J T, then Su C Sr. Any semigroup homomorphism X : Su n M(T)---+C can be extended to a semigroup homomorphism X : Su ---+C by declaring x( 11) = 0 for 11 r/:- M(T). Note that if 11,11' E Su,l1 + 11' E Su n M(T) = Su n ~ then 11, n' E ~ This implies that the above extension by zero is a weIl defined semigroup homomorphism. Thus we have an inclusion of the affine open set Uu(T) corresponding to the co ne ij E star(T) into Uu C ~ These inclusions patch up to give a weIl defined morphism V (T) ---+ X ~ which is 1 - 1. We regard this map as an inclusion.

Let (T ~ T,(T E ~ then a point X E Uu is in V(T) if and only if X(l1) = 0 for alln E Su such that 11 r/:- M(T), or equivalently, XU(x) = 0 for all 11 r/:- M ( T). This allows us to describe the ideal of V ( T) n U u in Uu as ttJUES",urt-rl.CXu. From this we see that V(T) is closed in Uu and is stable under the T action on ~ Hence Ou C V(T), as Xu E V(T).

N OW V (T) is the closure of the orbit Or since V (T) is a closed sub-


variety of X(ß) and contains OT as a dense subset. Moreover, as V(T) is stable under the T-action, it must be a union of T-orbits, namely, 07J for certain 'Tf E ß. We claim that V(T) = Ua>T Oa. We have shown that Oa C V(T) for all (j ~ T in ß. Suppose that x E V(T). Let 'Tf E ß be the smallest cone such that T C 'Tf and x E U 7J ( T ). Thus x : S7J ---t C is a semigroup homomorphism such that x (u) = 0 for all u 1:. M ( T). Suppose that u E S7J' -u 1:. Sw We claim that x(u) = O. If not, then clearly u E M(T) C ~ Now consider the cone (j = 'Tf n ~ Since u E ~ T C (j. The homomorphism x : S7J---tC extends to a unique semigroup homomorphism x : Sa = S7J + Z>o( -u)---tC which shows that x E U a ( T ). This contradicts our choice of 'Tf. Hence x ( u) = 0 if -u 1:. S7J' This implies that x E 07J'

Note that if T C T' E ß, then V(T') C V(T). Thus we have an order reversing correspondence between cones in T E ß and T-orbit closures V(T). If xE V(T) \ Ua>T V((j), then x 1:. Oa C V((j) for (j ;;;: T. Therefore x E OT as V(T) = U7J;T07J'

We summarise the above discussion as

Theorem 2.5.7. (i) Every orbit of X(ß) is of the form OT for some TE ß. (ii) Ua = UT<a OT' (iii) The orbZt closures V ( T) are in order reversing bijection with cones in ß. In fact V(T) = U 7J :::T 0w (iv) OT = V(T) \ Ua2;:T V((j). D

As an immediate consequence of the above theorem, we obtain

Corollary 2.5.8. V(T)nUa = 0 ifT is not aface of(j. V(T)nV(T') f= 0 if and only if there exists a (j E ß such that T and T' are faces of (j. D

Example 2.5.9. (i). Let N be a lattice with basis VI,'" ,Vn and let UI,'" ,Un denote the dual basis of M = H om( N, Z). Let (j denote the cone R:::OVI + ... + R:::ovn so that Ua = Spec( C[XI, ... ,XnJ) where Xi = X1Li • Let V = VI + ... + Vn , and let ß denote the fan in N whose n dimensional cones are: (ji,l ::; i ::; n, generated by V,VI,'" ,Vi-I,Vi+I,'" ,Vn· The identity map of N is a morphism 'P: ß---tßa , where ß a is the fan consisting of all faces of (j. The induced morphism 'P* : X(ß)---tX(ßa) = Ua is just the blow up of 0 in Ua ~ cn constructed in Example 2.5.6. In fact Uai = Spec(C[Sa;]) = Spec( C[Xi, XIXi-l,··· ,XnXi- 1 J). The morphism Uai ---tUi defined as


Xi H Yi, XjXi- 1 H lj,j =I i is an isomorphism of varieties. These

isomorphisms patch up to yield an isomorphism f : X(.6.)--+X. The composition p 0 f equals the morphism <p* under our identification of Ua with cn. The element Xa E Ua corresponds to 0 (since Xi(Xa) = XUi(xa) = xa(ud = 0). The inverse image of X a is just the orbit closure V(T) where T = R>ov. It follows from Example 2.3.8 that V(T) = X(star(T)) = pn-l, ~ it should be. (ii). More generally, let .6. be any fan in which we have a nonsingular cone a spanned by basis vectors VI,··· ,Vn of N. Denote by ..6. the fan obtained from .6. by replacing a by all faces of each of the cones ai, 1 ~ i ~ n. Then identity map of N is a morphism <p : ..6.--+.6. and the induced morphism <p* : X(..6.)--+X(.6.) is the (equivariant) blow up of X (.6.) at the T fixed point X a E Ua .

Proposition 2.5.10. Intersection of any two closed Tinvariant subvarieties in an affine toric variety is nonempty. Suppose that X (.6.) is an affine toric variety. Then.6. has a unique maximal cone a and X(.6.) = Ua .

Proof: Let X := X(.6.) be an irreducible normal affine toric variety with co ordinate ring A = C[X]. Thus A C Ao = C[M] and is M graded, that is A = EBuEBCXu for some finitely generated semigroup B C M which is saturated. If V is a nonempty Tinvariant irreducible closed subvariety of X, then the ideal I(V) C A of V is Tinvariant. It follows that I(V) is M-graded, that is, I(V) = EBuESCXu for a suitable subset S C B. Indeed S = {u E BlxulV = O}. In particular, if V and V' are any two closed nonempty Tinvariant subvarieties of X, then I(V n V') = EBuES"CXU , where S" C B is the saturated semigroup containing S U S'. If V n V' = 0, then I(V n V') = A and so 1 = Xo E I(V n V'). That is 0 E S U S'. It follows that there exists auE B such that u E S, -u rf. Sand -u E S', u rf. S'. Now X- u E A, XU E I(V) implies that X-uXu = Xo = 1 E I(V) since I(V) is an ideal. It follows that I(V) = A and V = 0, a contradiction. This shows that V n V' must be nonempty.

Now let .6. be a fan in N such that there exist more than one maximal co ne in.6.. Suppose a and T are two maximal dimensional cones in .6.. Then V (a) and V (T) are two nonempty disjoint Tinvariant closed subvarieties of X(.6.). It follows that X(.6.) is not affine. 0

Remark 2.5.11. Let X be an n dimensional normal, finite type complex algebraic variety. Assume that X contains a dense open subvariety

2.6. Resolution 01 singularities 61

Uo isomorphie to the algebraic torus T = (c*)n such that the multiplication action of T on Uo ~ Textends to an algebraic action T x X ~ X. Then there exists a fan ß in the lattice N = zn such that there exists a T = TN equivariant isomorphism X ~ We shall outline a proof of this fact. Let U be any affine open T stable subvariety of X. Since Uo is dense in X, and U is T stable it follows that Uo c U. Choose an isomorphism C[Uo] ~ C[XI, xlI,· .. ,Xn , X;;-I] = Ao. Since U is stable under the T action, C[U] is a T algebra. In particular, C[U] ~ EBuESCXu, where SCM:= Hom(N;Z) ~ zn. Here, for U = LUjej E M, XU

denotes the monomial xt 1 ••• ~ Since C[U] is aT-algebra, SeM is a semigroup. Since U is normal, C[U] is integrally closed. It follows that S is saturated, i.e., if ku E S, k E N, U E M, then U E S. Furthermore, since U is affine, C[U] is finitely generated as an algebra. Hence S is finitely generated as a semigroup. Now let a V denote the cone generated by S in MR. Clearly, a V is a rational polyhedral cone in M. Let a be the dual of a V in N. Then ais a rational polyhedral cone in N. Since U is n dimensional, it follows that a is strongly convex. It is a basic result, known as Sumihiro's theorem, that under our hypotheses, X is covered by finitely many T-stable affine open subvarieties. The collection of all cones a obtained by taking T stable affine open subvarieties of X forms a fan ß in N. For this fan, one has X ~ X(ß). We omit the details.

We conclude this section with the following remark.

Remark 2.5.12. Let X(ß) be a nonsingular toric variety and let TED.. Then V ( T) is also nonsingular .

Proof: Let a E ß be any cone that contains T so that (j E star(T). Let a = R:;;>OVI + ... + R?,OVk, and let T = R?OVI + ... R?ovr . Since X (ß) is nonsingular , VI, ... ,Vk is part of a basis for N. It follows that Vr +l + NTl · .. ,Vk + NT is part of a basis for N(T) = N /NTl where NT is the sublattice of N generated by N n ITI = ZVI + ... + Zvr . That is (j E star( T) is generated by part of a basis for N (T). It follows that V ( T) is nonsingular . 0

2.6 Resolution of singularities

A celebrated theorem of H.Hironaka says that any normal variety over an algebraically closed field of characteristic zero cau be desingularised. That is, if X is any such variety, there is proper birational morphism


Y ----+X with Y smooth. In particular, this would guarantee existence of desingularization for complex toric varieties. However, for toric varieties one obtains desingularizations Y ----+X such that Y is also a toric variety and the morphism is induced by a morphism of the corresponding fans. We shall give an explicit construction of such a desingularization in the case of toric surfaces.

Let a be a 2 dimensional scrap cone in N = Z2 and let D.cr denote the fan consisting of all faces of a. Suppose that VI and V2 are the first vectors in N along the edges Tl and T2 of a. If {VI, V2} is a basis for N, then Ucr is smooth. Suppose that {VI, V2} is not a basis for N. Then there exist lattice points in the triangle with vertices the vectors 0, VI, V2 other than the vertices themselves. Consider the convex polygon obtained as the convex hull of these lattice points along with VI, v2. Let ~ = VI, ~ ... ~ = V2 denote the successive vertices of the polygon. Let D. denote the fan consisting of 2 dimensional cones ai = ~ ~ + ~ 1 :::::; i < k, and their faces. Since the only lattice points in the triangle with vertices 0, ~ ~ are the vertices themselves, it follows that ai, 1 :::::; i < k, is nonsingular. Hence X(D.) is smooth. Now the identity map of N is a morphism 'P : D.----+D.cr of fans. Hence we obtain a morphism of toric varieties 'P* : X (D.)----+Ucr • This map is birational since 'P* induces the identity map of the torus TN . This morphism is proper by Theorem 2.5.5. Therefore 'P* is aresolution of singularities.

If D. is an arbitrary fan in N, then we do the above construction successively for each a E D. ~ is not nonsingular and obtain in a finite number of steps a fan D. ~ is nonsingular. Again the identity map of N is a morphism 'P : D. ----+ D.. 'fhe ind uced morphism 'P* : X(b.)----+X(D.) is a desingularization. This is because composition of proper (resp. birational) morphisms is proper (resp. birational).

Let N be of rank n ;:: 1. We call a fan D. in N a refinement of a fan D. in N if the identity map of N is morphism b.----+D. and 1b.1----+1D.1 is onto. We state the following theorem and refer the reader to §8, Ch. VI, [29] for its proof.

Theorem 2.6.1. Let D. be any fan in N ~ Then D. has a refinement D. which is nonsingular. The morphism 'P : D.----+D. induced by the identity map of N induces a desingularization 'P* : X(b.)----+X(D.). 0

2.7. Complete nonsingular toric surfaces 63

2.7 Complete nonsingular torie surfaees

Let b.. be a complete nonsingular fan in N = Z2. Then clearly b.. must have at least three 2 dimensional cones. Suppose it has exactly three 2 dimensional cones, then, by applying an automorphism of N if necessary, we can suppose that one of the cones is a = R"2üel + R"2üe2' Suppose that T E b.. is an edge which is not a face of a. Let v E T be the generator of the semigroup N n T. Write v = ael + be2, a, bE Z. Since ai = R>üei + R>üv E b.., i = 1,2 are cones, we must have a, b < 0.

- -

Since al is nonsingular , it follows that b = -1. Similarly, since a2 is nonsingular, we must have a = -1. Hence X(b..) == p2.

Assurne that b.. has at least four 2 dimensional cones. Thus we must have at least four edges in b... Let VI, V2, ... ,Vk be the first lattice points along the edges of b... Without loss of generality we may assurne that ai := R"2üVi + R"2üVi+1 is a co ne in b.. where Vk+l := VI. Since { Vi-I, vd and {Vi, Vi+ d are both bases for N, we must have Vi+ 1 = aivi-l + biVi, with ai = ±1, and bi E Z. Also, if Vi+l = -Vi-I, then a = -1, b = 0. Either Vi E R"2üVi+l + R"2üVi-l in which case ai = -1, bi > 0, or -Vi E R"2üVi+l + RVi-l in which case ai = -1, bi < 0. Thus the matrix of the transformation that maps. the ordered basis {Vi-I, Vi} to

the ordered basis {vz, vi+d is ~ ~ The numbers bz and the

vectors VI, V2 determine the fan b... Therefore the isomorphism type of X(b..) is completely determined by the numbers bi, 2 :S i :S k.

Now let 1 :S i,j :S k, i f. j. We claim that the following situation cannot arise: Vj = aVi + bVi+l, Vj+l = CVi + dVi+l, with a < 0, b > ° and C, d < 0. For, then the matrix of the transformation that maps the

ordered basis {Vi, Vi+ d to the ordered basis {v j , v j+ d namely (: ~ has determinant ad - bc ?:: 2, since a, b, c, d are all integers. But this is a contradiction since the determinant should be 1. Reversing the ordering of the Vi we see also that it is impossible that d < 0, b > ° and a < 0, b < ° hold simultaneously. Geometrically, this property translates into the following.

(*): The following situations cannot happen: (i) Vj is strictly between -Vi and Vi+l and simultaneously Vj+l is strict1y between -Vi and -Vi+l'

(ii) Vj+l is strictly between -Vi+1 and Vi and simultaneously Vj is strictly between -vi and -Vi+l'

Now suppose that k = 4. Since b.. is complete, both V3 and V4 cannot be in the same quadrant determined by V1 = e1, V2 = C2. Suppose that


V3 "# -v] and is in the seeond quadrant, then V4 cannot be in the interior of the third quadrant by the above observation. Suppose that V4 "# -V2· Then V4 is strictly between -V2 and -V3 whereas V5 = VI is strictly between -V3 and V2. This again eontradiets the above property (by labelling the vectors in the reverse order, V4, V3, V2, vd. This fore es V4 = -V2 = -e2. The resulting fan .6.(a) consists of edges R>OVl, R>OV2, R >o( -VI + aV2), R>o( -V2) for some integer a ::::: 0. When a ~ 0, the ~ toric v;'riety is the product Fo := pI x pI. When a "# 0, the corresponding torie variety Fa := X(.6.(a)) is known as the Hirzebruch surfaee. The first projection Z 2----rZ is a morphism of fans <p : .6.(a)----r.6.' := {R>oel,R>o(-eJ),O}. Also, the inclusion Ze2 C Z2 induces a ~ i : .6.ii := {R:::oe2,R:;::o(-e2),0}----r.6.. The indueed morphism of the toric varieties is, by Proposition 2.3.10, the projection <p* : Fa ----r pI = X (.6.') of a fibre bundle with fibre i* : pI = X(.6.")----rFa. Notice that Fa ~ F_a.

Theorem 2.7.1. If X (.6.) is a nonsingular complete toric surface, then X(.6.) == p2, pI X pI = Fo, or Fa for some a > 0, or X(.6.) is obtained by a sequence of equivariant blow ups at certain T fixed points starting with a p2 or a Hirzebruch surface Fa, a ::::: 0.

Proof: Let .6. be a nonsingular complete fan in Z2 having k ::::: 5 edges. We shall show that there exists an i such that Vi = vi - l + Vi+!. By Example 2.5.6 it would then follow that X(.6.) is the blow up of X(.6.') at the T fixed point X(j where .6.' is the complete fan having k - 1 edges generated by vI,··· ,Vi-I, Vi+l,··· ,Vk. The theorem thell follows by induction on k.

AllY half space H u = {v E R2 1 (u, v) ::::: O} C R2 eontains at most k - 1 vectors from the sequence VI, ... ,Vk . By relabelling the vectors if necessary, we may assurne that VI,· .. ,Vj is the longest sequence such that they all lie in the same half space of R 2. We claim that Vj = -VI. If not, Vj is strictly between -Vi and Vi+1 for any i < j - 1. Consider the vector Vj+ 1 . Our hypothesis implies that -VI is strictly between Vj and V)+1. Choose i < .i such that V)+I is strictly between -Vi and -Vi+ 1. If i < j - 1, then this contradicts (*) as Vj is strietly between Vi+l and -Vi. If i = j -1, then V)+I,··· ,Vk, VI,··· ,Vj-l alllie in a half plane. Our assumption on labelling of the vectors implies that j + 1 = k. Now Vk+l = VI is between -Vj and Vj-l, Vk is between -Vj-1 and -Vj. This again eontradicts (*). Therefore we must have Vj = -VI. Now if j = 3, then k = 4 contradicting our hypothesis that k ::::: 5. Therefore j ::::: 4.We shall show that for so me 1 < i < j, Vi = Vi-1 + Vi+l. Write

2.1. Complete nonsingular toric surJaces 65

Vr = -arVl + brV2, ar, br E Z,2 < r ::; j. Let Cr = ar + br. Note that ar , br 2 0, and Cr 2 2 for r < j. If C3 = 2, then V2 = VI + V3 and so we can take i = 2. So assume that C3 2 3. Since Cj = 1, there has to be an i < j such that Ci > CH 1 and Ci 2 Ci-I' Thus 2Ci > Ci-l + Ci+ 1. Since dVi = Vi-l + Vi+l for some integer d > 0 we have d· Ci = Ci-l + Ci+l. The only way this can happen is if d = 1 and so Vi = Vi-l + Vi+ 1. 0

Remark 2.7.2. As noted above, the Hirzebruch surface Fa is a pI bundIe over pI. In particuIar, Fa is a projective variety. Since blow-up of a projective variety at a point is again projective, it follows from the above theorem that any complete nonsingular toric surface is projective. In higher dimensions, however, there are nonsingular toric varieties which are compiete but not projective. See [32] for details.

We have seen that ~ is compietely determined by VI, V2 and the numbers b2,'" ,bk where biVi = vi-l + Vi+l. In particular the isomorphism type of ~ is determined compietely by the bi since there is an automorphism of the Iattice which maps VI, V2 to el, e2. The bi 's must satisfy the following conditions: Let b1 be the integer such that bl VI = Vk + V2.

Lemma 2.7.3. Let ~ be a nonsingular complete Jan in the rank 2 lattice N = Z2. With the above notations one has (i) b1 + ... + bk = 3k - 12 (ii) The product oJ the matrices

~ ~~ ... ~ ~~ = I d.

Conversely, iJ the bi satisJy conditions (i) and (ii) , then Jor any basis Vl,V2 oJ N, the complete Jan ~ in N with edges Vl,V2,'" ,Vk is nonsingular.

Proof: When k = 3, we have seen above that V3 = -VI - V2. Therefore, b1 = b2 = b3 = -1 and the equation (i) holds. When k = 4, after a cyclic relabelling of the Vi, we have shown that V4 = -V2, and V3 = -VI + aV2, where a is an integer. This implies that b2 = a, b3 = 0, b4 = -a, b1 = 0 and so (i) holds. Now assume that k 2 5 and that the equation (i) holds for all compiete nonsingular fans with fewer than k edges. By the above theorem, there exists a j such that Vj = Vj-l +Vj+l, where Vj-l, Vj, Vj+l all lie in the interior of a half space. Consider the compiete fan N whose edges are those in ~ except R?ovj. We have bj- 1 = Vj-2 + Vj =


Vj-2 + Vj+l + Vj-I· Hence (bj-I - l)Vj_1 Vj-2 + Vj+l. Similarly, (bj + 1 - l)Vj+1 = Vj-I + Vj+2. Therefore the llumbers ~ corresponding to the fan D.' are related to bi as folIows: The sequence ~ is obtained from the sequence bi by omitting bj = 1 and replacing bj-I and bj + 1

by bj_l = bj - I - 1, bj = bj + 1 - 1. Therefore L bi = L ~ + 3 = 3(k - 1) + 3 - 12 = 3k - 12.

The assertion (ii) follows from the observation that the composition <PI o· . ·0 <Pk of the transformation <Pi : N ~ N defined by Vi-I f--7 Vi, Vi f--7

Vi+l , 2 ~ i ~ k + 1 (with Vk+2 = V2) is just the identity map. As for the converse, the integers bi determille the vectors Vi , 3 ~ i ~ k having chosen a basis VI, V2 for N. Now condition (ii) ensures that for any i, Vi, Vi+l is a basis for N. We let D. be the collection of all cones ai = R?O'Vi + R?O'Vi+l, 1 ~ i ~ k and their faces. Condition (i) implies that the vectors 'go around the origin' exactly once. This implies that D. is a complete nonsingular fan. 0

In the following sections we study the topology of toric varieties.

2.8 Fundamental Group

In this section we describe the fundamental group of the toric variety X(D.) where D. is any fan in N. We take Xo = 1 E TN =: T to be the base point. A convenient base point for Olj C UIj will be XIj. However, the base point will be suppressed in our notation.

Lemma 2.8.1. Let a be any n dimensional cone zn N where n = rank(N). Then UIj is contractible.

Proof: Let V E lai n N be in the interior of a. Let H : UIj x [0, ~

be defined as folIows: for x E UIj, and 0 ~ t ~ 1, let H(x, t)(u) = t(u,v)x(u) Vu E SIj. Note that H(x, t) is a homomorphism of semigroup

~ and hence it may be regarded as an element of UIj. Since (u, v) ~ 0 for u E SIj' H is continuous. Clearly, H ( -, 1) = id, and H( - , 0) is the constant map x f--7 Xlj E UIj. This proves the lemma. 0

Note that we have the canonical isomorphism ?Tl (TN) = N explicitly given by mapping v E N to the homotopy dass of a v : [0, ~ = Hom(M, C*), av(t)(u) = exp(2?THt(u, v)) Vu E M.

Lemma 2.8.2. Let a be any k dimensional eone in N. Then Olj is a deformation retraet of UIj. In particular, the fundamental group of UIj is N(a) = N/NIJ ~ zn-k.

2.8. Fundamental Group 67

Proof: This follows from Example 2.3.9 since Ocr can be identified with TN(cr) and the fibre Uo- is contractible. However, we give an explicit deformation. Choose v E N in the relative interior of (J". Define H : Ucr x [0, 1]---+Ucr as folIows: For x E Ucr , and u E Scr let H(x, t)(u) = t(u,v)x(u), if t > 0,

{ ° if (u, v) > ° H(x, O)(u) = x(u) if (u, v) = 0.

Then H is a weIl defined continuous map and yields adeformation retraction to Ocr C Ucr as required. It follows that one has an isomorphism 7rl(Ocr) ~ 7rl(Ucr), Now Ocr being the orbit of Xcr by the TN action with isotropy the connected group TNa , one has an isomorphism 7rl(Ocr) ~ 7rl(TN)/7rl (TNa ) = N/Ncr = N(a).

This proves that 7rl(Ucr) ~ N(a). 0

Corollary 2.8.3. Let a be any serap eone in N. Then the inclusion map T C Ucr induees a surjeetion 7rl(T)---+7rl(Ucr),

Proof: By ~ above lemma, the inclusion Ocr---+Ucr induces an isomorphis m of fundamental groups. Ocr is the quotient of T by the isotropy TNa at Xcr . Since TNa is connected, the action map fcr : T---+Ocr defined by t f---t txcr induces a surjection of fundamental groups. We claim that fcr is homotopic to the inclusion map T C Ucr . To see this, let a : [0, 1]---+Ucr be any path such that a(O) = Xo E T c Ucr , a(1) = Xcr . (Here Xo E T = Uo is the unit element xo(u) = 1, Vu E M of T.) Then F : T x [0, 1]---+Ucr defined as F(t, s) = t.a(s) is the required homotopy. o

Corollary 2.8.4. Let ~ be any fan in N. Then the fundamental group of ~ is isomorphie to the group ~ = N/Nt:. where Nt:. C N is the sublattiee generated by UcrEt:. lai n N. In partieular if ~ has an n dimensional eone, then X ~ is simply eonneeted.

Proof: The sets Ucr , a E ~ forms an open cover for ~ and Uo = T c Ucr for any a ~ By a generalization of the Van Kampen theorem, and using the inclusion T C Ucr , we see that ~ is the free product of 7rl(Ucr) = N(a) amalgamated over the surjections 7rl(T) = N---+ 7rl (Ucr ) = N(a). It is easily seen that this group is nothing but N(!::::.). 0


Example 2.8.5. Let b. be the fan in Z2 eonsisting of

al = R 2o(el + 2e2), a2 = R20(2el + e2), and T = O.

Then 7fl (b.) is the free produet of N(at} = Z[e2l and N(a2) = Z[ell with amalgamations i1 : N = Zel + Ze2--tN(ad where e2 1---7 [e2], el 1---7

-2[e2l, and i2 : N --tN(a2) where el 1---7 red, and e2 1---7 -2[ell. Therefore, in 7fl(X(b.)), one has the relations -2[e2l = ided = i2(el) = [eIl and -2[ed = [e2l· Henee -4[e2l = -[e2l and so 3[e2l = O. Therefore [e2l = -2[e2l = red· It follows that 7fl(X(b.)) ~ Z/3Z.

Remark 2.8.6. It is known that if X is an irredueible normal eomplex variety and U a (Zariski) open subvariety of X, then the inclusion U C X induees a surjeetion of the fundamental groups. Sinee any torie variety is normal, it follows that the fundamental group of any torie variety is abelian.

2.9 The Euler characteristic

Sinee for any serap eone a in N, Ua is homotopy equivalent to Ga ~ TN(a) ~ (c*)n-k, where dim(a) = k, the integral eohomology ring of Ua

is isomorphie to the exterior algebra

Reeall that we have a eanonical isomorphism 7f1 (U a) ~ N (a). Hellee H*(Ua ; Z) = A*(M(a)).

Let b. be any fan in N and let a1,' .. , a m be the maximal dimensional eOlles ~ Thus the Uai form an open eovering of X(b.). One ean use the speetral sequellee assoeiated to this eovering which converges to the Cech eohomology of X(b.). The E l terms of this speetral sequenee is given by

Lemma 2.9.1. The Euler characteristic of X(b.) is equal to the number of n dimensional cones in b. where n = rank(N).

Proof: Sinee the speetral sequenee above converges to the cohomology of X(b.), the Euler charaeteristic X(X(b.)) of ~ equals

L L (-l)Qrank(Hq(UT ; Z)) = L X(UT ).

TEll O:S q:Sdim( T)

2.9. The Euler characteristic 69

If T is not n dimensional, then X(UT ) = 0 whereas if T is n dimensional then X(UT ) = 1 by 2.8.1 and 2.8.2. Hence ~ = m, the number of n dimensional cones in ~ 0

We shall condude this seetion with a description of the second cohomology of a toric variety. In §2.11 we shall study cohomology of nonsingular projective toric varieties.

Let ~ be a fan such that all maximal dimensional cones in it are n dimensional. (For example this holds if ~ is complete.) Each Ua is then contractible. Therefore ~ = 0 for q > 0, and the complex ~ is the cochain complex

The differentials in this complex are the alternating sum of the morphisms induced by indusions: I:I araI r---+ I:I I:jff.I( -1)kaIO"J, where J = IU {j} and k is the number of elements in I which are bigger than j. Note that this is just the cochain complex associated to the simplex whose vertices are the O"i. In particular the cochain complex is exact and so ~ = 0 for p > O. This allows us to describe the second cohomology of ~

Lemma 2.9.2. Let ~ be such that alt maximal dimensional cones are n dimensional. Then

Proof: As noted above ~ = 0 for p ~ 1, and so in particular, Ei'o = o. Also, we have Eg,2 = 0 since all maximal dimensional cones in ~ are n dimensional. Therefore, ~ Z) ~ E)x} ~ ~ ~ Ker(d1 :

~ The map d1 : E;,l = ttJi<jHl(Uainaj; Z)----+ ttJi<j<k

H 1 (Uainajak; Z) = Ei,l is the alternating sum of morphisms induced by indusions: I:l<l<m,li:i,j Elak where ak is the indusion Uainaj f---J

Uainajnak and El =--1 if i < l < j and is 1 otherwise. Since the fundamental group of UT is N(T), one sees that H1(UT ; Z) = Hom(N(T), Z) = M(T), and the lemma follows. 0

We shall interpret an element of the right hand side of the above description of the second cohomology as 1 - cocyde that defines a line bundle ~ The above lemma is then the assertion that the ehern dass map defines an isomorphism


under the hypothesis that all maximal dimensional eones in ß are n dimensional.

2.10 Line bundles

Any torie fibre bundle with fibre type C is a line bundle, Le, a loeally trivial (algebraie) bundle with fibre type C and structure group C*. Any line bundle over a torie variety is upto isomorphism a torie fibre bundle such that the fibre is C. We shall not prove this here (cf. [32]). In view of this fact one obtains a eomplete deseription of the Pieard group of a torie variety. We shalt assume throughout this section that ß is nonsingular and that alt maximal dimensional cones in ß are of dimension n. We shall denote the set of all n- dimensional eones in ß by ß max. We shall put a total ordering -< on the set ß max.

A (holomorphie) line bundle Cover a eomplex analytie manifold X is given by a data {Ua, fa,ß} where {Ua} forms an open eovering; the transition maps fa,ß : Ua n Uß----+C* = GL1(C) are eomplex analytie, and they satisfy the l-eoeycle eondition fa,ßfß'f = fa'flUa n Uß n Uf . Explieitly, the eorresponding line bundle has total spaee E = U Ua X

C / '" where Uß x C 3 (x, z) '" (x, fa,ß(x)z) E Ua x C for any x E

Ua nUß, Z E C. Eaeh Ua is a trivializing open set for the bundle C. As we shall see below, any torie line bundle over a nonsingular affine

torie variety is trivial. If C is a torie line bundle over a torie variety X, {Ua }aEßmax is therefore a eovering of trivializing open sets. The transition maps fa,T : Ua n UT----+C* being a unit in AanT C Ao, are neeessarily of the form {Xu" /XUa }, where U a , U T E M. X- Ua and X- u ,. are rational functions on Ua and UT respeetively. Sinee XU " /XUa is nowhere vanishing on Ua n Un we have U a - U T E M((J n T).

Conversely, any data {Ua}aEßmax with U T - U a E M((J n T),T,(J E

ßmax determines a line bundle C given by the patching data {Ua, fa,T = XU,,-Ua }. Explieitly, the total spaee of Cis obtained from UaEßmax Ua X

C by identifying

The action of T is given by t.(x, z) = (tx, X-Ua (t)z) for (x, z) E Ua xC. For U E Sa + U a, one has a loeal seetion ~ : Ua----+CIUa ~ Ua x C defined as x f---7 (x, XU - Ua (x)). Let XW E Aa. Then

2.10. Line bundles 71

= (x, XW(x)XU-Uo- (x)) = 8w +u (X).

Thus the Aa-module qUa, C) of all sections of Cover Ua is isomorphie to Aa XUo- under the map ~ t-+ Xu .

If ~ ~ are sections of Cover Ua , UT respectively, then they agree on Ua n UT precisely when

Hence we obtain XU- UT (x) = XU' -UT (.E) for all x E UT nU a. Hence U = u' and so U - U T E ST. Conversely, if U E M is Ruch that U - U a E Sa for all a E .b.max, then the family of local cross seetions ~ yield a global section 8 u : X --7 C such that Su I U a = ~

Since XUo-- UT is nowhere vanishing on Ua n Un the zero loeus of s ean be obtained by taking the union of the zero loci of the loeal seetions

Note that w hen U a = U E M for all a E .b. max, then the eorresponding line bundle C is trivial. Conversely, if C is isomorphie to a trivial bundle, then U a = U T for all a, T E .b. max. (Here we need the hypothesis that maximal dimensional eones in .b. are n dimensional.)

Let C and C' be two (toric) line bundles eorresponding to the data {ua }, ~ with Ua - U n ~ - ~ E M(a n T) for any a, TE .b.max .

The tensor prod uet C ® C' eorresponds to {Ua + ~ }. In partieular, C is isomorphie to C' if there exists auE M such that U + U a = ~ for all a E .b.max and the bundle C- 1 ~ CV is given by the data {-ua }.

We shall now explain how to obtain the data {ua } when the line bundle C is given as a torie fibre bundle in the sense of 2.3.10.

Let .b. be any (nonsingular) fan in N. Consider a torie fibre bundle C: .b.' = {R>ovo,O} C .6.--7.b. as in Proposition 2.3.10. We shall regard - - - p N as N E9 N' where, in the exact sequenee 0--7N' y N ~ N--70 defining the morphisms offans, p is the natural projeetion N E9N' --7N. We assume Vo is a generator of N', so that N' = Zvo. Denote by Li c .6. the "lift" of.b.. The projeetion N --7N is a morphism of fans Li--7.b. such that {O} c Li --7.b. is a torie fibre bundle with fibre C*. This is the principal C* bundle assoeiated to C.

Observe that if .b. eonsists of single maximal eone whieh is non~ say a, so that X(.b.) = Ua , then the same is true of .6. and

X(.b.) ~ Ua x C. In partieular any torie line bundle over Ua is trivial as remarked above.

Now let .b. be any nonsingular fan in whieh all maximal eones are n-dimensional. Let VI,··· ,Vd ENdenote the primitive veetors along

72 2. Toric Varief'ies

the edges ~ The edges of ~ are ~ generated by the primitive vectors VO,Vl - alvO,'" ,Vd - advO E N for some integers al,'" ,ad. Let (T E ~ be generated by ViI"" ,Vin • Denote by Uer E M the element defined by Vi r H- -air' 1 :s; r :s; n. The collection {uer } as (T varies over the n dimensional cones in ~ satisfies the condition that Her - UT vanishes on I(T n TI. Conversely, starting with any such collection { uer } of elements in M, one readily recovers 3. (and hence .6.). The line bundle so obtained is the same as L associated to the data {u".} described above.

Let L be a (toric) li ne bundle over X ~ Note that (u er - uT ) E

K er( EBer--<TEßmaxM((TnT) ----+EBer--<T--<77Eßl11ax M((TnTn'T])) = H 2 (X ~ Z)). If L is such that U er =: U is independent of the n dimensional cone (T in ~ then L is a trivialline bundle. To see this, note that, in this case, the morphism <Pu : N ----+ N = N EB N', V H- v - (u, v)vo is a morphism of fans

~ The composition N ~ N ~ N is the identity morphism of ~ Therefore, <Pu induces a cross section of the principal C* bundle associated to the line bundle L. Hence L is trivial.

Now the collection {uer } yields loeal trivializations

Uij = p-l(Uer ) <1>'; ) Uer x C

1 1 Uer

id ----7 Uer

induced by the isomorphism of fans ~ ~ x ~ defined by ~ fj----+N EB N', Vir + (Uer,ViJVo H- Vi r , 1 :s; r ~ n, and Vo H- Vo. Thus, <per(v) = v - (uer,V)vo. The semigroup So: c N is generated by Uir,'/.lOHer E M. Note that the semigroup morphism Ser EB Z?,ouo----+S(j induced by the morphism ~ ----+ ~ X ~ of fans is defined by Uir H- Ui r and Uo H- Uo - Uer E So:. It follows that the induced morphism Aer[Xol----+Ao: is a morphism of Aer algebras defined by XUQ =: Xo H- XoX- u". If T E ~ and if x E Uer n Un then an element (x, z) E UT X C is identified with the element ~ 0 (<p:)-l(x,z) E Uer x C. The Xo coordinate of this element can be computed to be XU,.-U" (x)z by tracing the image of Xo under the composition Aer[Xol----+Ao: C AO:ni----+AernT' This shows that the line bundle is associated to the data {uer }erEßmax.


Lemma 2.10.1. Let .6. be a nonsingular fan in N in which alt maximal dimensional cones are n = rank(N) dimensional. Then, wüh the above notations, (i) The ehern class map Cl : ~ Z), maps [,e] to (Ur - ~ E H 2(X(.6.; Z)) and is an isomorphis,,!!: of groups. (ii) Let Ci denote the line bundle where the edges of .6. are va, Vi -

va, Vj, 1 ~ j ~ d, j # i. Then the isomorphism classes of Ci generate Pic(X(.6.)). More precisely, C ~ ~ &;; ••• &;; ~

Prüüf: We shall only prove that [Cl f-T (Ur - 'Ua ) is an isomorphism of groups. We omit the proof that this is indeed the Chern class map.

Note that the second assertion follows from the first since both line bundles C and ~ &;; ..• &;; cad have the same first Chern classes. That the map [Cl f-T (Ur - Ua ~ is a weIl defined homomorphism follows from our above discussion. If [Cl maps to zero, then Ua = Ur for all a, T E .6.max . Hence the bundle is trivial. If a = (ua,r ~ is in H 2 (X (.6.); Z), we first show that there exists a family {ua } such that ua,r = Ur - Ua. Indeed, if'l] is the largest element of .6.max , then define u1) = O. For any element a E .6.max , define U a = -ua ,1). Then for any a -< T E .6. max, we have Ur - Ua = ua,U'1 - ur ,1) = ua,r since by projecting a to M(a n T n '1]) we obtain 0 = ua,r - ua,1) + ur,1)" This proves our claim. Now the bundle C, given by the data {ua ~ maps to the given element a, proving surjectivity. 0

Let Su : X ~ C be the global cross-section given by the family of local cross seetions ~ where U - U a E Sa for all a E _.6.max . It is straightforward to verify that the monomorphism N ~ N defined as v f-T v - (u, v)vo is a morphism of fans ~ The induced map

~ is just the cross seetion SU . In particular su is T(= TN) equivariant where T acts on X(.6.) via the ~ ~ ind uced by the above monomorphism N ~ N.

The local section sulUa ~ corresponds to the Aa-homomorphism AO' ~ Aa defined by the semigroup homomorphism SO' =

Ba + Z::::o(uo - ~ that maps (uo - ua) to U - Ua and is identity on Ba.

The space HO(X(.6.); C) of all glohal sections of Cis then the complex vector space spanned by the toric seetions {su}. By restricting these sections to any Ua it is easy to show that these sections are also Clinearly independent.

Denote by Pe the convex hull of the set {u E M I U - Ua E Ba Va E

.6.max }. (We allow the possibility that Pe is empty.) Note that Pe =


{U E MRlu - U(J" E a V Va E .6.max } = {u E MRI(u,v) ~ (u(J",v) Vv E

lai, a E .6.max }.

Summarising the above discussion, we have

Lemma 2.10.2. Let .6. be a nonsingular fan in N such that alt its maximal cones are n = rank(N) dimensional. Then HO(X(.6.); 1:) = EBuEIP.cnMI C Su . . The sections Su are T -equivariant and for each a E

.6. max, the section ~ I U (J" is given by

o

Suppose I: is an algebraic line bundle over a (smooth) complete variety X. Then it is a general fact that the space V := HO(X; 1:) of all global sections of X is finite dimensional. Suppose that I: is generated by sections, i.e., there exists (algebraic) sections so,' .. ,Sk forming a basis for HO(X; 1:) such that for each x E X, the elements Si(X), 0:::; i :::; k span the fibre over x. Thus one has a surjection of vector bundles X x V ---+1: where (x, 2::0<i<k ZiSi) f--t 2::1 <i<k ZiSi(X). Taking duals, one has an injection I:v --+X x V V of vector -bundles. Thus the fibres of LV are one dimensional vector subspaces of VV. This leads to a a morphism CPC : X ---+P(VV) defined by x f--t fibre of I:v over x. Recall that the line bundle I: is said to be very ample if CPc is an imbedding. I: is said to be ample if 1:0k is very ample for some positive integer k . Thus if I: is ample, then X is projective.

Let .6. be a nonsingular complete toric variety and let I: be a line bundle on X(.6.) given by the data {U(J"}(J"Eß where U(J" - UT E M(a n 7) Va,7 E .6. max. We shall obtain a criterion for I: to be very ample. Define a piecewise linear map 'ljJ := 'ljJc : NR---+R as follows: 'ljJ(v) = (u(J"'v) for v E lai, a E .6.max . Note that if v E lai n 171, then (U(J"' v) = (un v) as U(J" - UT E M(a n 7). Thus'ljJ is indeed well-defined.

Recall that a continuous function f : N R---+ R is called convex if f(av + (1- a)w) ~ af(v) + (1- a)f(w) for all v , w E NR and 0< a < l. We say that f is strongly convex (with respect to .6.) if it is convex and such that strict inequality holds in the above whenever v, w do not belong to the same n-dimensional cone of .6..

We state without proof the following proposition:


Proposition 2.10.3. Let .c be a line bundle over a smooth complete toric variety ~ Then.c is generated by sections if and only if 'l/Je is convex. The bundle .c is very ample if 'l/Je is strongly convex. Equivalently .c is very ample if and only if {u - uulu E Pd generates the semigroup Su for alt a E ~ max . 0

We refer the reader to §3.4, [32] for proof. We remark that if .c is ample then it is very ample.

Remark 2.10.4. When .c is very ample the vertices of the polytope P : = Pe are the U U , a E ~ max. The toric variety X ~ associated to the rational convex P as in §2.4 is then isomorphie to ~ See [32] for details.

Example 2.10.5. Fix i, 1 ::; i ::; d. Let .ci be the line bundle defined in Lemma 2.10.1. This bundle is associated to the data {uu }uEßmax

where U u E M is defined as Uu = 0 if Vi is not on an edge of a. If Vi is on an edge of a, then -Uu is defined to be the element in the dual basis with respect to the basis of N consisting of generators of a such that (-Uu , Vi) = +1. It is straight forward to check that 0 E Pe. Let So denote the corresponding global section. We claim that the zero locus Zo = {x E ~ = 0 E p-l(x)} of So is V(Td, where Ti E ~ denotes the edge R'20Vi. To see this, let "7 E ~ and x E UTJ = Homs,g(STJ' C). If x E V(Ti), then UTJ n Vh) i= 0 and so Ti is an edge of "7. We have -UTJ fI- M(Ti) as UTJ(Vi) = -1. It follows that x 0 SÖ(uo - uTJ) = x(O - UTJ) = 0 as x E Vh). Since "7 E ~ was arbitrary, we obtain that Vh) C Zo0 Now let x E Zo n UTJ with "7 E ~ We must show that x E Vh). If Ti is not an edge of "7, then UTJ = 0 and so x 0 s6(Uo - UTJ) = x(O) = 1. Hence x fI- Zo a contradiction. Therefore, Ti is an edge of"7 and so UTJ generates M / M (Ti). Now x E Zo implies that x( -uTJ) = x 0 so(uo - uTJ) = O. If w fI- M(Td, then writing w = -auTJ + u', with u' E STJ n Mh), a > 0, we see that x(w) = x(-uTJ)ax(u') = O. Hence x E V(Ti). We re mark that the multiplicity of Vh) as the zero locus of So is 1. Thus the first Chern class of .ci is the class dual to V(Td. Therefore, denoting the dual cohomology class of V(Td by [vh)], we have Cl (.cd = [vh)].

Remark 2.10.6. Let .c be a line bundle over ~ corresponding to the data {uu }uEßmax. Let T ~ Choose any a E ~ such that T < a and set UT := UU ' Note that the restrietion of UT to the space


(NT)R is independent of the ehoice of (T used in the definition of UT. The rest riet ion LIV(T) of L to the subvariety V(T) is (isomorphie to) the line bundle given by the data {uij} where uij := uTJ - U T E M(T) where i7 E star(T)max is the image of"l E ~~

2.11 Cohomology of torie varieties

Throughout this seetion we assume that X = ~ is a nonsingular eomplete torie variety. Weshall give a deseription of the integral (singular) cohomology of X assuming a certain eombinatorial property holds

~ This property is known to hold for all projeetive nonsingular varieties. (See [32].)

Denote by Tl, ... ,Td the edges ~ Let Vi, 1 ::; i ::; d, denote the first lattice veetor along the edge Ti. One has a (torie) line bundle Li

eorresponding to eaeh edge Ti as defined in 2.10.1. We denote the first Chern dass of this bundle by Ci E H 2 (X; Z). Thus [V ( Ti)] = Ci by 2.10.5.

Lemma 2.11.1. (i) Cil ... Cik = 0 in H*(X; Z) if Vii'··· ,Vik zs not contained in a cone of ~

(ii) Ll<i<d(u, Vi)Ci = 0 in H 2(X; Z) for alt U E M.

Proof: By Example 2.10.5, the line bundle Li has a section Si : X -----+Li whose zero loeus is Di := V(Ti). Consider the veetor bundle E = EBl::;r::;kLir and the seetion s : X -----+E given by x J----t (x, Sil (x),··· ,Sik (x)). This vanishes on Z = Dil n· .. n D ik . Sinee there does not exist any eone that eontains vi), ... ,Vik we have Z = f/J by 2.5.8. Henee the bundle E admits a nowhere vanishing seetion. It follows that its top Chern dass must vanish. That is, Cil ... Ci k = 0 in H*(X; Z). This proves (i).

Given any U E M, we obtain a torie line bundle L given by the data {U(T := U ~ Clearly, this bundle is trivial. Therefore its first Chern dass must be zero. Note that, writing -ai := U(Vi) = (U, Vi), we have L ~ 0Lfi by Lemma 2.10.1. Taking first Chern dass, we get 0 = Li aici = L -(u, Vi)Ci = Ll<i<d -(u, Vi)Ci in H 2(X; Z), establishing (ii). - - 0

Let (Tl, ... ,(Tm be a listing of maximum dimensional eones of ~ For 1 ::; i ::; m, let "li E ~ be the faee of (Ti obtained by interseeting with (Ti

those (Tj with j > i for whieh (Ti n (Tj is n - 1 dimensional. Thus "ll = 0 sinee ~ is eomplete and "lm = (Tm. When X = X(ß) is nonsingular projeetive, there exists a listing of the maximum dimensional eones sueh

2.11. Cohomology of toric varieties 77

that the following holds:

For a proof for this fact see [32]. Assume an ordering of the maximum dimensional cones has been

chosen so that (*) holds. We shall obtain a decomposition of X as a union of affine spaces. That is, we show that the variety X is "paved by affine spaces".

Lemma 2.11.2. (i) For each cone , E ß, there exists a unique i = i(,) such that 'f]i -:; , -:; (Ji. In fact i is the smallest integer such that (Ji contains ,. (ii) If, < " E ß, then i(,) < i(,').

Proof: Suppose that 'f]i -:; , -:; (Ji and 'f]j -:; , -:; (Jj with i < j. Then 'f]j -:; (Ji w hich contradicts (*). This proves the uniqueness of i (,). Let i be the smallest integer such that , -:; (Ji. Then, is the intersection of certain (n - l)-dimensional faces of (Ji. Thus, is the intersection with (Ji of certain (Jj which meets (Ji in an (n - 1 )-dimensional face. By our choice of i, we must have j > i. This shows that 'f]i -:; , and i = i(,).

Since , -:; " -:; (J i(-y' l' we must have i (,) -:; i (,') by the second statement of (i). This proves (ii). 0

Let Yi = V('f]d n UUi' and let Zi = Yi U··· U Ym, so that we have a decreasing filtration X = Zl ~ ... ~ Zm = {XUm }.

Lemma 2.11.3. Each Zi is a closed subvariety of X and Zi \ Zi+l = Yi ~ Cn - ki where ki = dimCf7d·

Proof: If Yi n Yj, i :I j is nonempty, then there exists a , E ß such that 0, C Yi n Yj. Hence 'f]i -:; , -:; (Ji and ru -:; , -:; (Jj. This contradicts Lemma 2.11.2(i). This shows that Zi \ Zi+l = Yj. To see that Zi is closed, note that the closure of Yi is the union of those 0,' such that , -:; " E ß for so me , E ß with 'f]i -:; , -:; (Ji. By Lemma 2.11.2(ii) it follows that for such a ,', there exists a j > i with 'f]j -:; " -:; (Jj and so 0,' C Yj C Zi. Hence Zi is closed.

Now Yj = V(rld n UUi is just the affine toric variety corresponding to the (n - ki) dimensional nonsingular cone jj E star('f]i) in N('f]d. It follows that Yi ~ Cn - ki • 0

The subvariety V('f]i) Y X(ß) defines an element [V(rli)] in the (singular) cohomology group H 2(n-k;l(X(ß); Z), where 'f]i is ki dimensional.


This is Poincare dual to the homology dass represented by the submanifold V(7Jd thought of as a cyde in ~ (Cf. 2.5.12.) When 7Ji is one dimensional, V(7Jd is a co dimension one subvariety in ~ and [V(7Jd] is the first Chern dass associated to the line bundle Cj where Vj is the generator of 7Ji. See Example 2.10.5.

More generally, if T E ~ is generated by ViI"" ,Vik' the bundle [ = EBl::;r::;kCir has a seetion which vanishes on the subvariety V(T) (see proof of 2.11.1). The "degree" of vanishing of this seetion on V ( T) is 1. Since [ has rank k and V(T) has dimension (n-k), the dual cohomology dass [V (T)] represents the top Chern dass of [, which is Cil ... Cik' (See [34], p. 413.) Thus [V(T)] = Cil ... Cik E ~ Z).

The following theorem is due to Danilov (§1O, [24]). See also Jurkiewicz [54]. The proof is also given in Ch. 5 of [32].

Theorem 2.11.4. Let ~ be a nonsingular fan in N satisfying the property (*) above. Then the [V (7Jd] E ~ Z), 1 s i S m, form a Z-basis for the integral cohomology of ~ D

Let X be any compact nonsingular complex variety. Recall that Hp,q(X) = Hq(X, f2);), where f2P denotes the sheaf of germs of holomorphic p forms on X. When X is compact, HP,q(X) are well-known to be finite dimensional. When X is Kähler, one has the isomorphisms H:ing(X; C) = H1eRham(X; C) ~ EBHP,k-p(X). If V is a dosed complex analytic subvariety (not necessarily smooth) of X, then the dual cohomology dass [V] is of type (p,p), where pis the (complex) codimension of V in X. In particular, if H* (X; C) is generated by dasses of the form [V], V being a complex analytic subvariety, then HP,q(X) = 0 for p :I q. In view of the above theorem, we obtain the following:

Corollary 2.11.5. Let ~ be a nonsingular projective variety. Then HP,q(X) = 0 for p :I q. D

In particular, note that HO,q(X) = Hq(X, Ox) = O. In fact, if C is any line bundle over any complete variety, which is generated by sections, then all its higher cohomologies vanish. We record this fact below and refer the reader to [32] for a proof.

Theorem 2.11.6. Let ~ be any complete fan in N and let C be any line bundle over X := ~ which is generated by global sections. Then Hq(X:C) = 0 for all q > O. D

2.12. The Riemann-Roch Theorem 79

Remark 2.11. 7. In order to keep the exposition at an elementary level we have avoided consideration of Chow groups A*(X). As is usual in algebraic geometry, the i-th Chern dasses of an algebraic vector bundle E is defined as a certain homomorphism cflg(E) : A*(X)---tA*_i(X). There is also a natural transformation A*( - )---tH*( -; Z) which doubles the degree. It turns out that in our situation, when X = X(,ö) is a nonsingular complete torie variety whieh satisfies condition (*) above, then A*(X) ~ H*(X; Z) and the algebraie geometrie Chern dass map cflg(E) then equals the map ß f-7 ci(E) n ß for ß E H*(X). The reader is referred to [32] for a furt her details on Chow groups of toric varieties.

2.12 The Riemann-Roch Theorem

Let X := X(,ö) be a nonsingular projective variety. Let VI,'" ,Vd E N denote the primitive lattice points along the edges of ,ö. Let Ci denote the line bundle with Chern dass Ci E H 2(X; Z) defined in 2.10.1. Denote by T the tangent bundle of X. Regarded as a complex vector bundle, its total Chern dass can be described in terms of the Ci as follows:

c(X) := c(T) = rr (1 + Ci). l<i<d

Recall that the total Chern dass of Xis, by definition, the total Chern dass of the tangent bundle of X. The Todd dass of a li ne bundle I:- with total Chern dass of 1 + c is defined to be td(l:-) = 1 + td1 (I:-) + td2(C) + ... = c/(1 - exp( -c)) = (1 - cj(2!) + c2 /(3!) - ... )-1 regarded as an element of the cohomology ring H*(X; Q) = tfJk?:.oHk(X; Q). For a veetor bundle E over X, the Todd dasses may be defined using the "splitting principle" as follows: Write the total Chern dass of E formally as a product I11<k<r(1 + Uk) where r = rank(E). Then, td(E) is defined to be I11<k<r uk7(f-exp( -Uk)). The expression on the right hand side, being asymmetrie function in the Uj, can be expressed as a function in the Chern classes of E. Sinee the cohomology of X vanishes beyond the (real) dimension of X, we see that td(E) is in fact a polynomial in the Chern dasses of E. The total Todd dass of X, which is the same thing as td(T), can be computed from the above formula for the total Chern class of X although d exceeds the rank of T:

td(X) = rr cd(1- exp(-cd)· l<i<d


In particular, we have tdo(X) = 1, tdl (X) = (1/2)cdX), td2 (X) = (1/12)(CI (X)2 + C2(X)), ete.

The ehern eharacter ch(t:.) of a line bundle t:. with total ehern dass 1 + c is defined to be exp(c) E H*(X; Q). For an arbitrary (eomplex) veetor bundle f, one sets ch(t:) = 2::j exp(uj), which is again a eertain polynomial in the ehern dasses of f. The ehern eharaeter of X is

2::1:;j:;d cJ / k!. The Hirzebrueh-Riemann-Roeh Theorem (HRR) says that if f is an

algebraie vector bundle over a nonsingular projeetive variety X, then

x(X, f) = /\ ch(f).td(X).

Here X(X, t:) is the Euler eharaeteristic

2)-1)j dimc(Hj(X,f)) j

and Ix ß, for a eohomology dass ß E H*(X; Q) is just the evaluation (ß, fJ, x) E Q of ß on the fundamental dass fJ, X of the manifold X (with respect to the orientation coming from the eomplex analytic struct.ure on X). Sinee X(X, f) is evidently an integer, the same has to be true of the right hand side which is, apriori, only a rational number. It has been proved by Baum, Fulton, and MaePherson that the HRR formula is valid for any complete variety, even in the ease the variety X is not smooth. See [31J. The ehern dasses, Todd dass, ete., of X have to be defined suitably. It turns out that as in the smooth ease they are eertain rational linear eombinations of [V(a)J, a ~ (These numbers are not unique beeause of the relations among the [V(a)J.) We refer the reader to [32] for details.

Example 2.12.1. Let X = ~ be any nonsingular complete torie variety. Then HO(X, Ox) ~ C and Hk(X, Ox) = 0 for k > O. (See Theorem 2.11.6.) On the other hand, ch(O) = 1. It follows from HRR that the Todd genus 0/ X equals 1: Ix td(X) = 1. This is also true in the case X is not smooth.

Let t:. be a torie line bundle over a nonsingular eomplete torie variety X = ~ and let Cl (t:.) = c. By Theorem 2.11.6, if t:. is generated by its global seetions, then Hk(X, t:.) vanishes for k > O. Thus, in this ease, X(X, t:.) = dimHO(X, t:.). In view of 2.10.2, the dimension of

2.12. The Riemann-Roch Theorem 81

HO(X,'c) is just the number of elements of the lattice M in the polytope Pe.. Furthermore, Pe.0r = rPe. := {rulu E Pd. By the (RRR), sinee Cl (,C®r) = rc, we obtain

IrPe. n MI = L (ljk!) I. rkcktdn_k(X), l::;k::;n X

Let M R be given the Lebesgue measure normalised so that the volurne of the parallelepiped spanned by a basis for M is 1. Then it is not hard to show that Vol(P), the volume of the polytope P in M R all of whose vertiees are in M is given by

Using the above formula for IrPe. n MI we get

Let a E ß. Note that, sinee X is smooth so is V(a) by 2.5.12. For the line bundle 'cW(a) (which is still generated by global sections), one has Vol(Pda)) = (j*(c)kjk!,fLF(a)), where j : V(a)-+X is the inclusion, k = dim(V(a)), and Pda) is the polytope in M(a) = aJ.. nM assoeiated to the line bundle 'cW(a). Pda) C M(a)R C MR ean be identified with the face of Pe. obtained by intersecting with Pe. the coset -Ua + aJ... (Note that M(a)R = aJ...) Finally, fLF(a) is the generator of H 2k(V(a); Q) ~ Q corresponding to the complex analytic submanifold V(a). Note that (j*(c)kjk!,fLF(a)) = (ckjk!,j*(fLF(a))) = (ckjk!,[V(a)] n fLX) = ((ckjk!) U [V(a)],fLx) = fx(ckjk!)[V(a)], that is,

Vol(Pda)) = Ix (ck jk!)[V(a)].

Now let X = X(ß) be a complete torie variety and let ,C be a line bundle over X which is generated by global section. Writing td(X) = LaELl aa[V(a)] for suitable rational numbers aa, we obtain from the above expression for Vol(Pda)) and the (RRR) formula the following expression for the number of lattice points in Pe.:

IPe. n MI = L aaVol(Pc(a)). aELlI


The above formula for the volume of the polytope Pe is valid for any li ne bundle generated by sections over any complete toric variety, even when it is not smooth. In fact, given any convex polytope P with (finite number of) vertices in M having the same dimension as rank of M, consider the toric variety X := ~ corresponding to the fan ~ := ~

constructed in §2.4. This is a complete toric variety. There exists a line bundle 12 on X obtained as follows: The vertices of the polytope P correspond n-dimensional cones in ~ where n = rank(M). If u E M is a vertex of P and (J the corresponding cone in ~ max, then we set ~

to be u. Then the data ~ ~ defines the bundle C. This bundle is generated by sections and the polytope Pe is just the polytope P that we started with. Applying our results to the bundle 12 over ~ yields a formula for the number of lattice points in P in terms of the volumes of its faces and the rational numbers ~

The numbers ~ occurring in the expression for the total Todd dass of X are not easy to compute. However when X is a toric surface one has

td(X) = 1 + (1/2)([V(Tdl + ... + [V(Td)]) + [xl,

where Ti are the edges of ~ and [xl E H 4(X; Q) ~ Q is the dass dual to the point variety x. This leads to the following result known as the Pick's Formula: Pick's Formula: For a convex polygon P with vertices in Z2,

IP n Z 21 = Area(P) + 1/2{Perimeter) + 1.

Here length of an edge of P is to be calculated with respect to the measure where length of the segment between two consecutive lattice points along the edge is 1.

2.13 The moment map

Let G = {t E T = Homsg(M,C) It(u)1 = 1 Vu E M}, and let T> = {t E TI t(u) > 0 Vu E M}. It is dear that G and T> are subgroups oE T. Denote by R+ the set of positive reals. One ~ isomorphisms G ~ (sl)n and T /G ~ T> ~ (R+)n ~ Rn, the latter isomorphism resulting from the ~ map t H log(t), tE R+.

More generally, let (J be any (scrap) co ne in N and let ~ the affine toric variety. Denote by ~ > C U the subspace consisting of those ,-points x E U = ~ C) such that x(u) ~ 0 for all u E ~ Note that ifT < (J, then UT > C ~

'- '-

2.13. The moment map 83

One has the semigroup homomorphism C----+R?o defined by z t----7 Izl which is a retraction of R>o c C. This induces a continuous retraction r(J" : U(J"----+U(J",? defined by (r(J"(x))(u) = Ix(u)1 for all U E S(J". If T < (Y,

then r (J" restricts to the retraction of Ur onto Ur,?.

Let X ~ be a T -toric variety associated to a fan ~ in N ~ zn. From the above discussion, it is clear that one obtains a subspace ~ = U(J"Ell. U(J",? of ~ Also one obtains a continuous retraction r : ~ ~ by setting r(x) = r(J"(x) for any (Y E ~ which contains x. Note that the retraction is constant on the G-orbits in

~ Hence one has a map r : ~ defined by r. We claim that r is a homeomorphism. To see this, first ass urne that ~ is complete so that ~ is compact. Note that the G-orbit through each x E ~ meets ~ c ~ at a unique point. Hence r is 1 - l. Clearly r is surjective. The space ~ is compact and Hausdorff, being the retract of the compact Hausdorff space X ~ Hence r is a homeomorphism in the case when ~ is compact. The general case follows from this and the naturality of r with respect to inclusion of toric subvarieties since any X ~ imbeds into a complete toric variety by Theorem 9.3, Ch. VI, [29].

When ~ is smooth, the space ~ is a "manifold with corners", that is, ~ is locally homeomorphic to an open subset of (R?o)n. We shall see that when Xis nonsingular and projective, the quotient XjG ~ ~ is homeomorphic to a convex polytope P c MR. Note that N = H om(Sl, G) = 1f1 (G) and N R just the Lie algebra Lie( G) of G and MR, the dual of Lie(G). The polytope P is in fact the image of a moment map J-L : X ----+Lie(G)* = MR defined below. The polytope P is rational and the toric variety X ~ determined by P as described in §2.4, is nothing but ~

Let X = ~ be a nonsingular projective toric variety and let C be a very ample T-line bundle over ~ Recall from Lemma 2.10.2 that the space HO(X; C) of global sections of C is spanned by the Tequivariant sections SU, U E P n M where P := Pe is a rational convex polytope in M. The very ample line bundle C gives rise to a projective imbedding <Pe : X ----+pk, where k + 1 = IP n MI = ~ C)). Enumerating elements of P n M as Uo, ... ,Uk, the projective imbedding <P := <Pe can be explicitly described when .c is given in terms of the data { u(J"} that defines .c as follows: We shall write Si to denote the section SUi' etc. Since C is generated by sections, given x EX, there exists a ui E P n M such that Si(X) i- o. If x E U(J"' then Si(X) = si(x) =


(X, XUi -Ua (x)) -:j; O. That is XUi -Ua (x) -:j; O. Any sj can now be expressed as a multiple of sf(x). This multiple is just XUj-Ui(x) = x(Uj - Ui). We have <jJ(x) = [x(uo - ud,'" ,X(Uk - Ui)J E p k. By an abuse of notation, we shall write <jJ(x) = [x(uo),'" ,X{l.lk)J in view of the fact that x(u - u') = x(u)jx(u') if u, u', u - u' are all in Su.

Definition 2.13.1. With the above notations the moment map fJ : X ~ MR is defined as

where <jJ(x) = [zo,'" ,ZkJ.

Since XU(t.x) = XU(t)XU(x) for any t E T, x E X, it is dear that fJ is constant on the G-orbits of X. Hence fJ defines a map Ti : ~

Theorem 2.13.2. Suppose X is a nonsingular projeetive torie variety and that C is an ample line bundle. Then the image 01 fJ is a eonvex polytope P and Ti : X? ~ P is a homeomorphism.

Proof: Recall from §2.1O that the polytope Pe is defined as {u E M R I (1.l, v) ~ (uu, v) \Iv E lai}. The Uu E Pt:, a E Llmax are the vertices of Pe. (See 2.10.4.) By Proposition 2.10.3, the condition that C is (very) ample implies that Su is generated by {u - l.lu I U E Pe n M}.

Let a E Llmax and let X u E Uu denote the T-fixed point. Recall that xu(l.l) = 1 if u = 0, xu(u) = 0 if u E Su is non-zero (since a is n- dimensional). In particular, X u EX?. Now for u E Pe n M, XU-Ua(xu) = xu(u - uu) = 0 if u -:j; l.lU' Clearly, xa(uu - l.lu) = 1. It follows that fJ(xu ) = U U '

Now let T be any cone of Ll. Suppose that al,'" ,ap are the ndimensional cones which contain the face T. Then, writing Ui for UUi' since Ui-Uj E M(ainaj), 1 ~ i -:j; j :S p, we see that Ui-Uj E T.l. Hence XT(Ui - ur) = 1, 1 ~ i :S p. It is not difficult to see that xT(u - ur) = 0 for all U E Pe n M not in the face PT of Pe spanned by the vertices Ul,'" ,up . For any tE T?, the element t.xT EX? satisfies the condition that for U E Pe n M, t.xT(u - ur) > 0 if and only if U E PT n M. Thus Ti(t.xT) is a convex linear combination of the elements U E PT n M. The proof is completed by showing that for any TELl, the map Ti restricted to the T?-orbit of xT is areal analytic isomorphism onto the relative interior of PT' See Appendix on Convexity, in §4.2, [32J. 0

Chapter 3

Torus actions on manifolds

3.1 Introduction

The theory of torie varieties is a bridge between algebraie geometry and eombinatories, and we will explain in this note that it is possible to develop the theory from a topologieal point of view to some extent. In short, our aim is to constmct a bridge between topology and eombinatories.

A. tarie variety of dimension Tl is anormal eomplex algebraie variety with an action of (C*)n having a dellse orbit, where C* = C\{O}. Affine spaee cn and a eomplex projeetive spaee cpn with standard (C*)n_ aetions are typical examples of torie varieties, and other torie varieties may be viewed as a generalization of these examples. Although the eondition that the action of (c*)n have a dense orbit is very restrietive, torie varieties are abundant. The basic theory of torie varieties was established around early 70's by Demazure, Mumford ete., and MiyakeOda, see [32], [69], [24] and Chapter 2 in this volume. It says that

(1) there is a one-to-one eorrespondenee between torie varieties and eombinatorial objects ealled fans,

(2) a eompaet (nonsingular) torie variety together with an equivariant ample eomplex line bundle defines a moment map whose image is a lattiee eonvex polytope.

After the basic theory was established, many interesting applieations to eombinatorics (especially to eonvex polytopes) have been found. A simple but intriguing applieation is Piek's Formula. It says that if P is a lattice polygon (i.e., a polygon in IR2 with vertices in :f}) like figures

86 3. Torus actions on manifolds

shown below, then

1 the area of P = ~ - ~ - 1

where ~ denotes the number of lattice points in X.

• ~ l:-:J

Pick's Formula ean be proved by an elementary method, but when P is eonvex, it ean also be proved using the correspondenee (2) above. However, the formula holds even if P is not eonvex (see the right figure above) and this non-eonvex ease is not eovered by the theory of torie varieties. This awkwardness suggests existenee of a theory which aHows us to prove Pick's Formula in fuH generality, and we will find that such a theory does exist in topology. To be more preeise, our geometrical objeet is a torus manifold whieh is a smooth manifold of dimension 2n with an effeetive action of a eompact torus T = (SI)n (plus some orientation data). A compaet nonsingular torie variety with restrieted T-aetion provides an example of a torus manifold. Tools we use are purely topologieal and a main one is equivariant eohomology. Beeause of this, OUf argument works not only for eompact nonsingular torie varieties (with restricted T -actions) but also for torus manifolds. Interestingly, new (or more general) eombinatorial objeets appear in this extended eontext, i.e., fans are replaeed by multi-fans and eonvex polytopes are replaeed by multi-polytopes.

As is weH known, the eorrespondenee (2) above is generalized to eompaet sympleetie manifolds with torus aetions admitting moment maps. A sympleetic manifold M is a smooth manifold of even dimension, say 2n, with a nondegenerate closed two form w. Here w is said to be nondegenerate ifthe 2n-form wn is nowhere zero on M. A eompaet nonsingular torie variety together with an ample T-line bundle L provides an exampIe of a symplectic manifold with an action of T admitting a moment

3.1. Introduction 87

map, the two form being the first Chern form of L. In our case, M is a torus manifold and the complex li ne bundle L is arbitrary, so that the first Chern form of L is often degenerate although it is closed. Therefore, a torus manifold together with a complex T-line bundle does not necessarily provide an example of a symplectic manifold while it provides an example of a pre-symplectic manifold, that is a smooth manifold of even dimension with a (possibly degenerate) closed two form.

Although our theory brings us new insights into combinatorics, it is misguided that ours is a complete generalization of the theory of toric varieties. There are two defects in our theory for the moment. One is that we treat only compact and smooth manifolds while there are noncompact toric varieties and toric varieties with singularities. Among those singularities, quotient singularities are mild and accessible from a topological point of view. In fact, a toric variety with quotient singularities is an object studied in topology, which is called an orbifold (or a V-manifold). Orbifolds are close to smooth manifolds, and many results on smooth manifolds hold for orbifolds with a little modification. However, it would be hard to treat toric varieties with tough singularities from a topological point of view. The other defect is that the correspondence (1) above is not one-to-one in our case. We can associate a multi-fan with a torus manifold, but the correspondence is not injective. Furthermore, we do not know what types of multi-fans arise from torus manifolds. It is a fundamental problem in our theory to characterize multi-fans obtained from torus manifolds.

It is possible to develop the theory for torus manifolds as mentioned above (see [44]), but in this note we restrict our concern to so-called toric manifolds (or quasitoric manifold)l introduced by Davis-Januskiewicz ([25]) so that our arguments become simpler and clearer.

We are concerned with torus actions, but there are some developments for the non-abelian analogue. For instance, a non-abelian analogue of a toric variety is a spherical variety (see [17]) and the theory is developing, and there is a certain convexity theorem for actions of compact non-abelian groups on symplectic manifolds (see [59]). There is also a work on non-abelian analogue of (quasi)toric manifolds (see [75]). But these non-abelian analogues seem to be not yet fullyexploited.

1 The terminology "toric manifold" is used for a (complete) non-singular toric variety in algebraic geometry, so the terminology "quasitoric manifold" is used in the book [18] by Buchstaber-Panov.


3.2 Equivariant cohomology

Throughout this note T will be an n-dimensional compact torus group (sl)n and a smooth manifold with a smooth action of T will be called a T-manifold.

A main tool to study actions of T on manifolds is equivariant cohomology. In this section we review its definition and properties. All homology and cohomology groups are taken with Z coefficients unless otherwise stated.

Universal principal T -bundles

Let ET ----7 BT be a universal principal T-bundle, where ET is a contractible topological space with a free T-action. Usually we understand that T is acting on ET from the right. Such a bundle is unique up to homotopy, and there is an explicit description of ET and BT which we shall explain. Let s2m-l be the unit sphere of Cm and consider a free (right) action of 9 E SI C C by

The orbit space s2m-l / SI is a complex projective space Cpm-l. The inductive limits of natural inclusions

are respectively SOO = ~ s2m-l and Cpoo = U;;;=1 Cpm-l . The space SOO is contractible and has a free SI-action whose orbit space is Cpoo . Therefore SOO ----7 Cpoo is a universal principal SI-bundle. Its n-fold cartesian product is a universal principal T-bundle. Since H*(CpOO) is a polynomial ring over Z in one variable of cohomological degree two and BT = (Cpoo)n, H*(BT) is a polynomial ring over Z in n variables of co homologie al degree two.

Equivariant cohomology

Let M be a closed T-manifold. The balanced product ET XT M (here, the action of gE Ton (z, x) E ET x M is given by (zg-l, gx) and

3.2. Equivariant cohomology 89

ET XT M is the orbit space of the T-action on ET x M) is denoted by MT. We set

and caIl it the (q-th) equivariant cohomology of the T-manifold M. Equivariant cohomology was introduced by A. Borel around 60's

([13]). He used it to reprove the famous Smith fixed point theorem. Around 70's it was extensively used by Bredon (see Chapter VII of [14]), Hsiang brothers (see [51]) and others to study actions on appropriate spaces. In 80's Atiyah-Bott [6], Kirwan [58] and others realized that equivariant cohomology fits weIl to the study of actions on symplectic manifolds.

As is weIl known, Hq(M; IR) is isomorphie to the de Rham cohomology of M defined using forms on M. Although MT is not a manifold, there is a de Rham cohomology version of equivariant cohomology, see [6], [39].

If the T-action on M is trivial, then MT = BT x M; so HT(M) ~ H*(BT) 0 H*(M) by Künneth formula if H*(M) has no torsion. In partieular,

HT(pt) = H*(BT).

The projection 1[ from MT = ET xTM onto the first factor ET/T = BT defines a fiber bundle with fiber M:

~ ~ (2.1)

This fiber bundle is a product if the T-action is trivial, but twisted in general unless the action is trivial. One can expect that HT(M) reflects the twist (in other words, the T-action on M) to some extent. Through 1[*: H*(BT) -+ HT(M), one can view HT(M) as a module over H*(BT). The Serre spectral sequence of the fiber bundle is useful to cOlnpute the cohomology of the total space, that is HT(M). Generally speaking, it is not so easy to compute HT(M) but there is a very simple situation.

Lemma 3.2.1. 11 Hodd(M) = 0 and H*(M) has no torsion, then

(1) HT(M) ~ H*(BT) 0 H*(M) as H*(BT)-module,

(2) the retriction map ~ HT(M) -+ H*(M) is surjective and its kernel is the ideal generated by 1[* (H>o (BT) ).


Proof. The E2-terms ~ of the Serre spectral sequence of the fiber bundle (2.1) are given by

Since HOdd(BT) = Hodd(M) = 0, the spectral sequence collapses. Moreover, since both H* (BT) and H* (M) have no torsion, all the E2 terms have no torsion. The lemma follows from these facts. 0

The lemma above says timt HT(M) as H*(BT)-module does not reflect the T-action on M (satisfying the conditions in the lemma). In fact, its module structure over H*(BT) is the same as that of the trivial action. However HT(M) has more structures. It is a ring under cup product, furthermore the ring structure together with the H*(BT)module structure makes HT(M) as an algebra over H*(BT). These ring and algebra structures cannot be read off from the spectral sequence argument. We will see later that these structures on HT(M) reflect the T-action on M pretty weIl when M is a toric manifold.

Localization Theorem

A benefit of equivariant cohomology is that a global invariant lying in the equivariant cohomology HT(M) can often be determined by local data around the T-fixed point set MT by virtue of the following localization theorem (see Chapter 1 in this volume or p.40 of [51]).

Theorem 3.2.2. [Localization Theorem} Let 8 = H*(BT)\{O}. Then the localized restrietion homomorphism

is an isomorphism, where 8-1 N for an 8-module N consists of alt fractions {m/ s I m E N, 8 E 8} with identification mI/ 81 = m2/82 if and only if 881m2 = 882ml for some 8 E 8.

Corollary 3.2.3. If Hodd(M) = 0 and H*(M) has no torsion, then the restrietion map HT(M) --t HT(MT ) is injective.

Proof. If Hodd(M) = 0 and H*(M) has no torsion, then HT(M) is free as a module over H*(BT) by Lemma 3.2.1. Therefore the natural map from HT(M) to 8-1 HT(M) is injective, and the corollary follows from the localization theorem. 0

3.2. Equivariant cohomology 91

The restriction map H:;'(M) -+ H:;'(MT) is far from surjective in general but there is a nice description of the image under certain conditions, see [36], Chapter 11 of [39] or [40].

Equivariant characterictic classes

There are three characteristic dasses for vector bundles. They are Euler dass, Chern dass and Pontrjagin dass. Let us review them briefly, see [65] for details.

Let 7r : E -+ M be a vector bundle.

(1) The Euler dass e(E) is defined for an oriented vector bundle E, and lies in Hq(M) where q is the fiber dimension of E. If the orient at ion on E is reversed, then the Euler dass changes the sign.

(2) The (total) Chern dass c(E) = 1 + Cl (E) + c2(E) + ... is defined for a complex vector bundle E, and the k- th Chern dass Ck (E) lies in H 2k(M). If r is the complex fiber dimension of E, then ck(E) = 0 for k > rand the top Chern dass cr(E) agrees with the Euler dass of the underlying oriented real vector bundle of E.

If Tm is a Hopf line bundle over a complex projective space Cpm (i.e., the fiber over a complex line f E Cpm consists of vectors in f), then Cl (Tm) is a generator of H 2 (cpm ).

(3) The (total) Pontrj agin dass p( E) = 1 + PI (E) + P2 (E) + . .. is defined for areal vector bundle E, and the k-th Pontrjagin dass Pk(E) lies in H 4k (M). In fact, Pk(E) is defined to be (-1)kC2k (E 0 C).

Using these characteristic dasses, one can define equivariant characteristic dasses like we defined equivariant cohomology using ordinary cohomology. Suppose that 7r : E -+ M is a T-vector bundle, Le. T acts on both E and M, 7r is T-equivariant, and 9 E T maps a fiber 7r- I (x) over x E M to 7r- 1(gx) linearly (and isomorphically) for any x. The T-vector bundle produces a vector bundle

1 XT 7r : ET = ET XT E -+ MT = ET xT M

of the same fiber dimension as E. Additional structures on E such as orientation and complex structure

will be inherited to ET. The equivariant Euler dass of an oriented Tvector bundle E, denoted eT (E), is defined to be the ordinary Euler dass of ET :

eT (E) := e(ET) E Hj,(M)


where q is the fiber dimension of E. An oriented T-representation space V can be viewed as an oriented T-vector bundle over a point, so eT (V) lies in Hj,(pt) = Hq(BT) where q = dirn V.

The equivariant Chern (resp. Pontrjagin) class of a complex (resp. real) T -vector bundle E -+ M will be defined in a similar fashion :

cT(E) := C(ET) and pT(E):= p(ET ).

When E is the tangent bundle TM of M, eT ( T M), cT ( T M) and pT ( T M) are often abbreviated as eT(M), cT(M) and pT(M) respectively.

Equivariant Gysin homomorphism

Let X and Y be closed oriented T-manifolds. For an equivariant continuous map 1 : X -+ Y, an H*(BT)-module homomorphism called an equivariant Gysin homomorphism

1! : Hj,(X) -+ Hj,+T (Y)

is defined, where r = dirn Y - dirn X, and has the same properties as ordinary Gysin homomorphism (see Chapter 1 in this volume or [56]). Here are those properties.

Property 1. Let h : Y -+ Z be an equivariant map between closed oriented T-manifolds. Then (h 0 1)! = h! 01!.

Property 2. f!(uJ*(v)) = 1!(u)v for u E Hr(X) and v E Hr(Y).

Property 3. Let Xl and X 2 be closed oriented T-submanifolds of X which intersect transversally. For the following diagram of inclusion maps

Xl nX2 i' Xl ---+

j1 i1 X 2 ~ X

we have an identity i; 0 j* = i* 0 K In particular, if Xl and X 2 have no intersection, then i* 0 j[ = O.

Property 4. If X is a closed oriented T-submanifold of Y and f: X -+ Y is the inclusion, then

1* 1!(1) = eT (v)

where 1 E ~ is the identity and v denotes the normal T-vector bundle of X in Y with orient at ion induced from those of X and Y.

3.3. Representations 01 a torus 93

3.3 Representations of a torus

We review a basic representation theory of the n-dimensional torus group T.

Complex representations 01 a torus

A complex one-dimensional representation of T is a homomorphism from T to GL(l, C) = IC*. Since T is compact, the image of the homomorphism lies in the maximal compact subgroup of IC*, that is, SI. Denote by Hom(T, SI) the set of homomorphisms from T to SI, in other words, the set of complex one-dimensional representations. It forms an abelian group under the multiplication on the target space SI.

Lemma 3.3.1. Any eomplex one-dimensional representation 01 T = (SI)n is given by

n

(tl,'" ,tn) E T -+ II ~ E SI k=l

with some n-tuple (mI, ... ,mn) E 71P. Moreover, different n-tuples produee nonisomorphie eomplex one-dimensional T -representations. In partieular, Hom(T, SI) is isomorphie to 71P.

We shall interpret Hom(T, SI) in terms of topology. For 1 E Hom{T, SI) let Vf be the associated complex one-dimensional T-representation space. Since ef{Vf) = eT(Vf) is an element of H 2 (BT), we obtain a map

ll1 : Hom(T, SI) -+ H 2 (BT)

which sends 1 to ef{Vf).

Lemma 3.3.2. ll1 is an isomorphism.

Proof. The case where n = 1, i.e., T = SI. In this case f: T = SI -+ SI is apower map, say a k power map. As remarked before, we may take SOO -+ Cpoo as the universal Sl-bundle. Consider the following commutative diagram

s2m-l X VI -----t s2m-l X Cm

1 1

94 3. Torus actions on rnani/olds

Here vertical maps are projections and the upper horizontal map is given by (ZI, ... , Zm, v) --+ (ZI, ... ,Zm, ~ ... , ~ where v E Vj = <C. Since all the maps in the above diagram are T-equivariant and the upper horizontal map is linear and injective on each fiber, the diagram induces a vector bundle map

s2m-l XT Vj --+ s2m-l XT Cm

1 1 cpm-l Cpm-l.

The image of the upper horizontal map is the k-fold tensor product (,m)k of the Hopf li ne bundle Im over Cpm-l. Therefore the first Chern class of the left line bundle over cpm-l is k times a generator of H 2(cpm-l). Taking m --+ 00, one sees that ci (Vj) is k times a generator of H 2(BS1 ).

The case where n is arbitrary. For i = 1, ... , n, let Ii: SI --+ SI be the restriction of / to the i-th factor of T = (sl)n. Then Vj is the tensor product of Vji 's over C, and according to this decomposition, the line bundle (Vj)T --+ BT decomposes into the tensor product of pullback of line bundles (VjJsl --+ BS1 by projection Pi: BT = (Bs 1)n --+ BS1

on i-th factor. It follows from naturality of characteristic classes that cf(Vj) = ~ pi(cf (VjJ), and this together with the observation for the case n = 1 above implies the lemma. 0

Since T is abelian, any complex representation of T decomposes into a direct sum of complex one-dimensional representations uniquely; so complex representations of T are completely understood.

Circle subgroups 0/ a torus

A dual version of the isomorphism W in Lemma 3.3.2 is

WV : Hom(SI, T) ~ H2(BT).

In fact, the map WV is defined as follows. Consider the pairing

Hom(T, SI) x Hom(SI, T) --+ Hom(SI, SI) ~ Z

defined by composition of maps. It is nondegenerate, so we have natural isomorphisms

Hom(SI, T) ~ Hom(Hom(T, SI), Z) ~ Hom(H2(BT), Z) ~ H2(BT).

The composition of these isomorphisms is our Wv. The following would be clear.

3.3. Representations of a torus 95

Lemma 3.3.3. For u E H 2(BT) and v E H2(BT), let XU E Hom(T, SI) and Av E Hom(Sl, T) be the corresponding elements via 1]1 and 1]Iv respectively. Then

The image of a nontrivial homomorphism from SI to T is a circle subgroup of T, but two such homomorphisms may determine the same circle subgroup of T. Through 1]1 V , one sees

Lemma 3.3.4. Two nonzero elements Al, A2 in H 2(BT) determine the same circle subgroup ofT if and only if they span the same "line" through the origin in H2(BT), in other words, if and only if m1A1 = m2A2 for some nonzero 'integers m1, m2·

Orbit spaces of T -representation spaces

Lemma 3.3.5. Let V be a faithful real T-representation space. Then

dirn VT + 2dimT ~ dirn V,

in particular, 2 dirn T ~ dirn V.

Proof. Take aT-invariant inner product on V. Then V decomposes into the direct sum of V T and its (orthogonal) complement (VT).l. Since a nontrivial irreducible real T-representation is two-dimensional and obtained as realification of a complex one-dimensional representation, one may think of (VT).l as a direct sum of complex one-dimensional T-representation spaces. Let Xl, ... ,Xk be their characters, where k = dimc(VT).l. The faithfulness of the T-representation space V is equiva

lent to the homomorphism ~ Xi : T -+ (Sl)k being injective. Therefore dimT ~ k. Since dirn V = dirn V T + 2k, the lemma follows. 0

The orbit space of a T-representation space V is not necessarily a manifold (with boundary), but it is in the following extreme case.

Lemma 3.3.6. If V is a faithful real T -representation space and 2 dirn T =

dirn V, then the orbit space V/T is homeomorphic to (IR+)n where IR+ denotes the set of non-negative real numbers.


Proof. Since 2 dirn T = dirn V, V T = {O} by Lemma 3.3.5 and V may be viewed as a direct sum of n complex one-dimensional Trepresentation spaces with characters Xi 'So The homomorphism TI?=1 Xi : T ---+ (sl)n is injective as remarked in the proof of Lemma 3.3.5, but since T and (sl)n have the same dimension, the homomorrphism must be an isomorphism. This implies that the orbit space V /T is homeomorphic to the orbit space of the standard linear action of (SI)n on Cn defined by

(ZI, ... ,Zn) ---+ (gI ZI, ... ,gnZn)

where,(ZI,'" ,Zn) E Cn and (gI, ... ,gn) E (SI)n. The latter orbit space is clearly (ffi.+)n, so the lemma is proven. 0

3.4 Torie manifolds

Henceforth the T -action on M is assumed to be effective unless otherwise stated. We begin with

Lemma 3.4.1. 1f MT =I- </J, then dimMT + 2dimT ~ dimM, in particular, 2 dirn T ~ dirn M, where dirn MT denotes the maximum 01 the dimensions 01 connected components in MT.

Proof. Let x be a point of a connected component in MT of the maximum dimension. Since the action is effective, the tangential representation space TxM is faithful; so dirn TxMT + 2 dirn T ~ dirn TxM by Lemma 3.3.5. Since dirn TxMT = dimMT and dimTxM = dimM, the lemma follows. 0

Locally toric manifolds

Motivated by Lemmas 3.4.1 and 3.3.6, we are led to the following notion.

Definition 3.4.2. A T -manifold M of dimension 2n is called locally toric if any point x of M has an invariant open neighborhood which is equivariantly dijjeomorphic to an open invariant neighborhood of some faithful real T -representation space V of dimension 2n.

The codimension of a point y E (lR+)n is defined to be the number of zeros in the co ordinate of y. A Hausdorff space Xis said to be a manifold

3.4. Toric manifolds 97

with corners of dimension n if it is given a set of pairs {(UA, 1/;A)} which satisfies the following conditions:

(1) {UA} is an open covering of X,

(2) 1/;A : UA -+ (IR+)n is a homeomorphism onto the image which is open in (IR+)n,

(3) if UA n Up, i= </J, then 1/;p, 01/;-;1 1/;A(UA n Up,) -+ 1/;p,(UA n Up,) preserves codimensions.

The local coordinate map 1/;A in (2) above is not required to be surjective. Since (IR+)n contains an open set homeomorphic to IRn, an ordinary manifold can be viewed as a manifold with corners (although it has no corner). Similarly a manifold with boundary is also a manifold with corners.

The following lemma is immediate from Lemma 3.3.6.

Lemma 3.4.3. If M is a locally toric manifold of dimension 2n, then the orbit space M /T is an n-dimensional manifold with corners.

Toric manifolds

A convex polytope P is the convex hull of a finite set of points in IRn .

We may assume that the points are in a general position (i.e., they are not in an affine hyperplane ) because if they are, then we may think of them as points in IRn - 1. A (proper) face of P is an intersection with an affine hyperplane in IRn such that P sits in a half-space divided by the hyperplane. The dimension of a face is the dimension of the minimal affine space which contains the face. A face of dimension n - 1 is called a facet of P and a face of dimension 0 is called a vertex. We say that P is simple if each vertex of P lies on exactly n facets. A simple convex polytope is a manifold with corners.

Example 3.4.4. Among five regular convex polytopes in IR3 , a tetrahedron, a hexahedron( =cube), and a dodecahedron are simple, and the others (i.e., an octahedron and an icosahedorn) are not simple. Another example of simple convex polytope is an n-simplex. Note that a tetrahedron is a 3-simplex.

Definition 3.4.5. Following Davis-Januszkiewicz [25j, we call a locally toric manifold M a toric manifold if the orbit space M /T is (homeomorphic to) a simple convex polytope as manifold with corners.


The following two are typical examples of torie manifolds whieh the reader should keep in mind. They are eompaet nonsingular torie varieties with restrieted T-aetions.

Example 3.4.6. Let Sl act on S2 = Cp1 as rotations fixing the north and south poles. Then the n-fold eartesian product (s2)n has an action of T = (Sl) n. One easily checks that this is a torie manifold. The orbit spaee is homeomorphic to [-1, 1 Jn and the T -fixed point set eonsists of 2n points.

Example 3.4.7. A eomplex projeetive spaee Cpn with an action of (91, ... ,9n) E T = (Sl)n defined by

where [Zl,"" zn+d E Cpn is the homogeneous eoordinate, is a torie manifold. A map from Cpn to IRn+ 1 defined by

induees a homeomorphism from Cpn /T to the standard n-simplex.

A eompact nonsingular torie variety X with restricted T-action is loeally torie and its T-orbit spaee is often homeomorphic to a simple eonvex polytope as manifold with eorners2 , for instanee this is the ease when X is projeetive; so it provides an example of a torie manifold in our sense. But there are many torie manifolds whieh do not arise from nonsingular torie varieties as we will see later.

The following example shows that a loeally torie manifold is not neeessarily a torie manifold.

Example 3.4.8. Consider

n

s2n = {( Zl, ... , Zn, x) E Cn X IR 1 L 1 Zi 12 + x2 = 1} i=l

with the action of T = (Sl)n defined by

2There is a compact nonsingular toric variety whose T-orbit space is not homeomorphic to a simple convex polytope as manifold with corners, see [21J.

3.4. Toric manifolds 99

One can see that s2n with this T-action is a locally toric manifold. The orbit space s2n /T is not homeomorphic to any simple convex polytope as a manifold with corners if n 2: 2 (but is homeomorphic to an n-ball as a topological space). Therefore the T-manifold s2n is not a toric manifold3 if n 2: 2.

Characteristic submanifolds

Let M be a toric manifold.

Definition 3.4.9. A connected codimension two submanifold of M is called characteristic if it is left fixed pointwise under some circle subgroup ofT.

Let AM be an index set pararnetrizing characteristic subrnanifolds of M, i.e., a characteristic subrnanifold of M will be denoted by Mi (i E

AM)' For a subset 1 of AM the intersection niEIMi will be abbreviated as MI. Observe that

(1) the T-action is free on M\ UiEAM Mi ,

(2) if MI # 1jJ, then Mi'S (i E 1) intersect transversally. (Hence dirnMI = 2(n - 111) and 111 :<:::: n, where 111 denotes the cardinality of 1.)

We will denote the circle subgroup which fixes Mi pointwise by Ti, and the subtorus generated by Ti (i E AM) by TI. Note that dirn TI = 111 and TI fixes MI pointwise.

Example 3.4.10. Consider Cpn with the T-action in Exarnple 3.4.7. One checks that the subrnanifold (cpn)i defined by Zi = 0 is characteristic, so ACpn = {I, ... ,n + I}. The circle subgroup Ti which fixes (cpn)i pointwise is the subgroup {(I, ... ,gi, .. " 1) I gi E SI} for i :<:::: n and the diagonal circle subgroup {(g, ... , g) I 9 E SI} for i = n + 1.

Excercise. For a positive integer k, consider a Hirzebruch surface

W(k) := {(la, b], [x, y, z] E Cpl X Cp2 I aky = bkx}

with the action of (gI, g2) E (SI)2 = T defined by

([a,b],[x,y,z]) ~ ~

3But this is a torus manifold, see [44].


(1) Check that W(k) is a toric manifold.

(2) Find four characteristic submanifolds W(k)i and the corresponding circle subgroups Ti.

(3) Find four T-fixed points and the tangential T-representations at them.

Construction of toric manifolds

The orbit space MIT of a toric manifold M is a simple convex polytope by definition. Note that MJ/T is a codimension 111 face of MIT.

Now we reverse a gear. Namely we start with a simple convex polytope P and construct a toric manifold with P as the orbit space. To do this we need an extra datum defined below. Let Ap denote an index set parametrizing facets of P , i.e., a facet of P will be denoted by Pi for i E Ap . For a sub set 1 of Ap, we denote by PI the intersection niEIPi . Note that 111 :::; n if PI i- <p. Suppose that we are given a map F: Ap --7 H2 (BT) such that

if PI i- <p, then {F(i) li E I} extends to a Z-basis of H2(BT).

The map F is called a characteristic map and provides a necessary information to construct a toric manifold with P as the orbit space. The construction goes as folIows. When PI i- <p, let TI be the 111-dimensional subtorus generated by circle subgroups of T determined by F(i) 's (i E I). (Remember that a nonzero element in H2(BT) determines a circle subgroup of T .) We define an equivalence relation on T x P: (91,pd ,...., (92,p2) if and only if PI = P2 and 9"1192 E TI where 1 is a sub set of Ap such that PI contains the point PI = P2 in its relative interior. One checks that the quotient space T x PI ,...." denoted by M(P, F), is a (smooth) manifold and the action of T by left translations descends to an action of Ton M(P, F), so that M(P, F) is the desired toric manifold with P as the orbit space.

The projection onto the second factor induces a map

q : M(P, F) --7 P.

Since the characteristic submanifolds of M (P, F) are given by q-l (Pi) (i E Ap), we may assume that Ap = AM(p,F) , and then q-l(PI) =

3.5. Equivariant cohomology of toric manifolds 101

M(P, Fh for any subset I c Ap. In particular, T-fixed points in M(P, F) correspond to vertices in P via the map q. Observe that the maximal subtorus which fixes M(P, Fh pointwise is TI generated by circle subgroups corresponding to F(i)'s (i EI).

Example 3.4.11. Take n = 2 and let P be a d-gon, that is a twodimensional convex polytope with d sides. The facets of P are the sides. We number them as PI, P 2 , ... ,Pd in such a way that Pi-l and Pi are adjacent for i = 1, ... ,d, where Po = Pd. Let VI, . .. ,vd be a sequence of elements in H2(BT) ~ 71,2 such that each successive pair Vi-l and Vi

is a basis of H2(BT) for i = 1, ... , d, where Vo = Vd. We then define F(i) = Vi for each i.

3.5 Equivariant eohomology of torie manifolds

Throughout this section, M is a toric manifold.

Cohomology of toric manifolds

Denote by Pi E H 2 (M) the Poincare dual of a characteristic submanifold Mi of M.

Lemma 3.5.1. A toric manifold M has a cell decomposition with even dimensional cells. In particular, M is simply connected (hence, orientable), Hodd(M) = 0 and H*(M) has no torision. Moreover, H*(M) is generated by Pi 's as a ring.

Proof. Take a height function q; on P = M /T such that q; takes distinct values on vertices of P, and decompose M using q; into cells as follows. We orient each edge of P so that q; decreases along it. For each vertex V of P, let m(v) denote the number of incident edges which point towards v (so that n - m(v) edges point away). Let Fv be the smallest face of P which contains the inward pointing edges incident to v. Clearly, dimFv = m(v). We delete from Fv all faces not incident to v. The

resulting space Fv is homeomorphic to ~ Let ev be the inverse image

of Fv by the projection map q : M ~ P. Since Fv is homeomorphic to


~ ev is homeomorphic to Cm(v) = IR2m(v) , see Lemma 3.3.6. The set {ev I v : vetices of P} gives a cell decomposition of M. Since each cell is even dimensional, this proves the former assertion of the lemma.

The closure of a cell of dimension less than 2n is an intersection of characteristic submanifolds Mi'S because q-I(Pi) = Mi and Fv is an interseetion of Pi's. This means that H*(M) is generated by f-li'S as a ring, proving the latter assertion of the lemma. D

There are many relations among (products of) f-li's, which will be described at the end of this section.

Simplicial complex r M

The T-action is free on M\ UiEA M Mi, so one can expect that characteristic properties of the action are concentrated on (neighborhoods of) the Mi's. Using Mi's, we define a combinatorial invariant

It describes which Mi 's intersect. Clearly r M is an abstract simplicial complex of dimension n - 1. Since MI =I </J if and only if (M /T)J =I </J, r M is determined by the orbit space M /T.

Exarnple 3.5.2. For M = Cpn in Example 3.4.10 rM consists of all proper subsets of AM = {I, ... , n + I}; so the geometrie realization of r M is the boundary of the standard n-simplex, which is homeomorphic to sn-I.

If I E rM, then it is a dimension 111 - 1 face of rM while (M/T)J is a co dimension 111 - 1 face of ä( M /T). This implies that r M is the "dual complex" of ä(M/T).

H:;'(M) and face ring

We shall see that the simplicial complex r M is closely related to the ring structure of H:;'(M). Since M is orientable and Mi is a fixed point component under a circle subgroup, Mi is also orientable. We choose orient at ions on M and Mi 's, and fix them. Then an equivariant Gysin homomorphism is defined for the inclusion map h : Mi -t M:


We set ~ := fi!(l) E Hj,(M)

where 1 E ~ is the identity. For a subset I of AM, we abbreviate ITiEI ~ as ~ Geometrically, ~ is the Poincare dual of Mi in equivariant cohomology, so ~ is the Poincare dual of MI. Therefore ~ must be zero whenever MI is empty, i.e., whenever I tf. f M . It turns out that there is no other relation in HT (M). N amely we have

Theorem 3.5.3. [see [25},[62}} Let n be a polynomial ring in variables Xi'S (i E AM) and let X be the ideal in n generated by monomials XI'S

for I tf. fM. Then the map " nix -+ HT(M) sending Xi to ~ is a ring isomorphism.

The ring nix is determined by f M and called the face ring (or Stanley-Reisner ring) of the simplicial complex fM. (See [77].) We derive a combinatorial invariant from fM. For 0 ~ k ~ n - 1, let fk(M) be the number of k-simplices in f M . Define a polynomial WM(t) of degree n by

n-l

W M(t) = (t - l)n + :L fdM)(t _1)n-l-k k=O

and let hk(M) for 0 ~ k ~ n be the coefficient of tn- k in W M(t), i.e.,

n

WM(t) = Lhk(M)tn- k. k=O

Excercise. Prove that WM(t2 ) = ~~ bq(M)tq, where bq(M) denotes the q-th Betti number of M. (Therefore hk(M) = rankH2k (M).)

Excercise. Prove that the Euler-Poincare characteristic

2n

:L(-l)qbq(M) q=O

of M agrees with the number of vertices in MIT.

104 3. Torus aetions on manifolds

Equivariant Pontrjagin dasses of torie manifolds

For x E MT, denote by I(x) the maximal subset of AM such that x E M1(x)' Note that II(x)1 = n. By definition ei = f!(1) where fi : Mi ---+ M is the inclusion. Its restriciton to x, denoted by eilx, vanishes unless x E Mi (in other words, unless i E I(x)), which follows from Property 3 of equivariant Gysin homomorphism stated in Section 3.2.

in H:;'(M).

Proof. By Corollary 3.2.3 and Lemma 3.5.1, it suffices to prove that the restrictions of the both sides at the identity above to any point x E MT coincide. The restriction of the left hand side to x is pT (TxM) while that ofthe right hand side is ITiEI(x) (1 +ellx). Thus it suffices to prove that

pT(TXM) = II (1 + ellx). (5.1) iEI(x)

Since Mi's (i E I(x)) intersect transversally at x, we have

TxM = EB vilx (5.2) iEI(x)

as T-representation spaces, where vilx denotes the fiber over x of the normal bumlle Vi of Mi in M. Taking the equivariant Pontrjagin classes on both sides, we obtain

pT(TXM) = II pT(Vilx). iEI(x)

Here pT(Vilx) is equal to the restriction of pT(vd to x, and since Vi is of dimension 2, we have

pT(vd = 1 + pf(vd = 1 + eT(vd 2 = 1 + ft(a),

where the second identity follows from Corollary 15.8 of [65] and the third identity follows from Property 4 of equivariant Gysin homomorphism stated in Section 3.2. Since g(el)lx = ellx, these identities prove the desired identity (5.1). D

Corollary 3.5.5. Let Mi be the Poineare dual of the homology dass in H 2n - 2 (M) represented by Mi. Then

p(M) = II (1 + Mr) in H*(M). iEA M


Proof. Restrict the identity in Theorem 3.5.4 to H*(M). Since ei restriets to Mi, the corollary follows. 0

Example 3.5.6. Let M be epn as in Example 3.4.10. Then AM {1, ... , n + 1} and each Mi is equal to a generator M of H2 (epn) up to sign. It follows that

which is a well-known formula (see §15 of [65]).

The algebra structure of Hr(M)

We have ~ that the ring structure of Hr(M) is determined by fM. Since fM is determined by the orbit space MIT, the ring structure of Hr(M) is determined by MIT and has nothing to do with a characteristic map F. However, Hr(M) has a finer structure than the ring structure, that is, an algebra structure over H* (BT) through 1[* :

H*(BT) ---t Hr(M). It turns out thg.,t the characteristic map appears in the algebra structure.

We have to investigate 1[* to know the algebra structure of HT(M) over H* (BT). To see 1[*, it suffices to see the image of elements in H2 (BT) through 1[* because H* (BT) is generated by elements in H 2 (BT) as a ring.

Lemma 3.5.7. There is a unique element Vi E H 2(BT) for each i E AM such that

1[*(u) = L (U,Vi)ei in Hf(M) for any u E H 2 (BT) iEÄM

where (, denotes the natural pairing between cohomology and homol-ogy.

Proof. Since Hj,(M) is additively generated by ei'S by Theorem 3.5.3, 1[* (u) can be expressed uniquely as

1[*(u) = L Vi(U)ei

iEÄM

with integers Vi (u) depending on u. We view Vi (u) as a function on H 2(BT). Clearly the function is linear, so it defines an element of Hom(H2(BT), Z) = H 2 (BT), that is our desired Vi. 0


Example 3.5.8. Consider Cpn with the T-action in Example 3.4.10. We give Cpn and characteristic submanifolds (cpn)i orienta-tions induced from the complex structures. Then Vi = (0, ... ,1, ... 0) (1 sits in the i-th place) for i = 1, ... , n and Vn +l = (-1, ... , -1) when H2(BT) is identified with 71P in an appropriate way.

Lemma 3.5.9. For any x E MT, the set ~ I i E I(x)} is a Z-basis of H 2(BT) and the set {Vi li E I(x)} is its dual basis.

Proof. Look at the identity (5.2). The tangential T -representation space TxM is faithful and dirn TxM = 2 dirn T. Therefore, the weights of the direct factors Vilx, which are ~ = eT(vilx), form a basis H 2(BT). This proves the former statement of the lemma.

Take u = ~ for j E I(x) at the identity in Lemma 3.5.7 and restriet it to x. It reduces to

~ = L ~ ~ iEI(x)

because ~ = 0 unless i E I(x). This implies the latter statement of the lemma. 0 Excercise. Prove that the circle subgroup determined by Vi is Ti which fixes Mi pointwise.

Define FM: AM -+ H 2 (BT)

by F( i) = vi for i E AM. Lemma 3.5.9 teIls us that FM is a characteristic map.

Excercise. Observe that M is equivariantly homeomorphic (or diffeomorphic) to M(M/T, FM).

If M = M(P, F) with a charcterictic map F : Ap -+ H2(BT), then AM = Ap and F(i) = FM(i) for each i E Ap = AM up to sign. The sign depends on the choice of an orientation on lvh One can always choose an orientation on Mi so that F(i) = FM(i) because FM(i) = Vi changes into -Vi if we reverse the orientation on Mi·

3.6. Unitary toric manifolds and multi-fans 107

Cohomology of toric manifolds revisited

Since HOdd(M) = 0 and H*(M) has no torision by Lemma 3.5.1, the restriction map L*: H:r(M) -+ H*(M) is surjective and its kernel is the ideal generated by 7r*(H>O(BT)) by Lemma 3.2.1, where 7r: MT -+ BT is the projection. Since H>O(BT) is generated by elements in H 2 (BT), it follows from Theorem 3.5.3 and Lemma 3.5.7 that

Theorem 3.5.10. [[25}} Let R be the polynomial ring in variables Xi 's (i E Atvt) as before and let i be the ideal in R generated by alt

(1) monomials XI (I rt- r M),

(2) 2::iEAM (u, Vi)Xi for u E H 2(BT).

Then the map : Rli -+ H*(M) sending Xi to ILi is a ring isomorphism.

Excercise. Check that H* (Cpn) is a truncated polynomial ring Z[J1,J/(J1,n+l) using Theorem 3.5.10, where J1, E H 2 (cpn) is a generator.

3.6 Unitary torie manifolds and multi-fans

Unitary toric manifolds

As observed in the last section, orientations on M and Mi 's are necessary to define the characteristic map FM. When M is a compact non singular toric variety, M and Mi 's are complex manifolds; so they have canonical orientations. However, our toric manifold M does not necessarily come from a nonsingular toric variety and there is no canonical choice of orientations on M and Mi 'so

To get around this, we impose a complex structure on the tangent bundle TM of M, more generally, on the stable tangent bundle of M. Such a structure is respectively called an almost complex structure or a unitary structure. The unitary structure is sometimes called a weakly almost complex structure. A complex manifold is an almost complex manifold but the converse is not true. An almost complex manifold is a unitary manifold but the converse is again not true. The sphere s2n admits a unitary structure for any n but admits an almost complex structure only when n = 1 or 3.

108 3. Torus actions on manifoids

Definition 3.6.1. We call a toric man'ifoid M a unitary toric manifoid if TM E9 ~ for some k is endowed with a compiex structure preserved by the T -action, where ~ denotes the product bundie M x ]R2k -+ M. When k = 0, a unitary toric manifoid is especially called an almost compiex toric manifoid 4 .

Henceforth M will denote a unitary toric manifold unless otherwise stated. Then TM E9 ~ is oriented as a complex vector bundle and ~ is oriented in the usual way. These orientations determine an orientation on TM.

Excercise. Let H be a subgroup of T. Prove that each connected component of M H admits a unitary structure and its normal bundle in M becomes a complex T-vector bundle with the complex structure induced from the one on TM E9 ~

By the excercise above, each Mi is unitary, so it has an orientation induced from the unitary structure. Thus M and Mi's have canonical orientations. The tangential T-representation space TxM at x E MT is a complex representation space since it is the nontrivial part of a complex T-representation (TM E9 ~ In fact , we have

TxM = L llilx as complex T-representation spaces. iEI(x)

We define the equivariant Chern dass cT (M) or the Chern dass c(M) of M to be ~ or ~ respectively. The same argument as in Theorem 3.5.4 and CoroUary 3.5.5 proves

Theorem 3.6.2. cT (M) = ITiEAM (1 + ei) and c(M) = ITiEAA.f (1 + fJ.i) .

Todd genus

The Todd dass of M, denoted by T(M), is a polynomial of Chern dasses ci(M) over rational numbers associated with the formal power series

z Z z2 ---=1+-+-+ .... 1 - e- Z 2 12

4The terminology "unitary (or almost eomplex) torie manifold" is used in [62] for a rat her more general family of unitary manifolds with T-aetions, and "unitary torus manifold" is used in [44] instead of "unitary torie manifold".

3.6. Unitary torie manifolds and multi-fans 109

Since c(M) = D(1 + J-ld, we have

2

T(M) = II J-li . = II(l + J-li + J-li + ... ). 1 - e-M, 2 12

Excercise. Check that T(M) = l+!cdM)+ 112(Cl(M)2+ C2 (M))+ ....

The number obtained by evaluating T(M) on the fundamental dass [M] of M is ealled the Todd genus of M and denoted by T[M]. Apparently, the Todd genus T[M] is a rational number, but it is aetually an integer. (See [47] and [8] for more details and its history.)

Example 3.6.3. Let M = CP2. We know that c(CP2 ) = (1+J-l)3 where J-l E H 2 (CP2 ) is a generator. Therefore

T( 2) ( J-l )3 (1 J-l J-l2 )3 1 3 2 CP = 1 _ e-M = + 2 + 12 + . . . = + 2 J-l + J-l + ... ,

The Todd genus of any eompact nonsingular torie variety is known to be one. But the Todd genera of unitary torie manifolds take any integers while those of almost eomplex toric manifolds must be positive as seen later and aetually take any positive integers, see [62].

Multi-fans

Before we introduce the notion of multi-fan, we make aremark on orientations. Remember that TxM at x E MT is a eomplex Trepresentation spaee. Therefore TxM has two orientations: one indueed from the eomplex strueture and the other from the global orientation on TM. They may not eoineide although they coineide whenever M is almost complex. Let 1 be an element of r M with 111 = n. Then MI eonsists of one T-fixed point, and we define

sgnM(I) = +1 or - 1

according as the two orientations at MI eoineide or not. As remarked above, sgnT(I) = +1 for all 1 E r Af with 111 = n if M is an almost complex torie manifold, in partieular, if M is a eompact nonsingular torie variety.


We have observed that the simplicial complex fM together with the elements Vi E H 2(BT) (i E AM) determine the algebra structure of H:r(M) over H*(BT). We combine these two data Cu and Vi'S as follows. To each 1 E fM we associate a cone LVI spanned by Vi'S (i E 1) in an n-dimensional vector space H2(BT; IR):

LVI := {L biVi E H2(BT; IPl) I bi ~ 0 for all i E I}. iEI

Here is a reason why we form cones.

Lemma 3.6.4. Let x E MT. By Lemma 3.5.7 any element V E H 2 (BT) can be written as V = I:iEI(x) biVi with integers k We view TxM as an

SI-representation through the homomorphism Av : SI -+ T corresponding to V. Then the weights of the SI-representation are alt positive if and only if alt the bi 's are positive.

Proof. We know that TxM = I:iEI(x) vilx. Using the notation in

Lemma 3.3.3, we may write TxM = I:iEI(x) ~ since eT(vilx) = ~ The weight of ~ restricted to SI via the homomorphism Av is given by ~ v) by Lemma 3.3.3, which is equal to bi by Lemma 3.5.9. This proves the lemma. 0

Now we corne to the notion of multi-fan.

Definition 3.6.5. The collection of cones LVI indexed by 1 E f M together with the sign function sgnM attached to maximal cones (i. e. ndimensional cones) is called the multi-fan of M and denoted by ßM.

Since Vi corresponds to Mi, LVI corresponds to MI for 1 E fM. In particular, a maximal cone LVI with 111 = n corresponds to a T-fixed point.

Excercise. Prove that the Euler-Poincare characteristic of M is equal to the number of maximal cones in the multi-fan ßM, that is equal to the number of T-fixed points.

When M is a compact nonsingular toric variety, the multi-fan ßM

agrees with the (ordinary) fan (with sgnM(I) = +1 for all 1 E fM with 111 = n) known in the theory of toric varieties. In this case cones in ßM

intersect at only their proper faces, so they do not overlap. However, unless M is a compact nonsingular toric variety, cones in ßM may overlap. The degree of the overlap is related to the Todd genus of M as the following theorem shows.

3.6. Unitary toric manifolds and multi-fans 111

Theorem 3.6.6. [[62]] Let v E H 2(BT; lR) be generic, t.e., v does not sit in any hyperplane spanned by Vi 'So Then

T[M] = :L sgnM(I)· I:vELv[

In particular, if M is an almost complex toric manifold, then T[ M] equals the number of maximal cones which contain v (hence T[M] > 0).

Proof. We may assurne that v is integral, i.e., v E H 2(BT). Since v is generic, the fixed point set of the restrieted action of the SI-subgroup Av (SI) agrees with MT. Then the Kosniowski formula for unitary manifolds ([46]) teIls us that T[M] equals the sum of sgnM(I) over 1 with 111 = n such that the weights occuring in TxM, where x = MI, viewed as the representation of Av (SI) are aIl positive. This together with Lemma 3.6.4 proves the theorem. D

The Todd genus of a compact nonsingular toric variety is one and sgn(I) = +1 for any 1 E r M with 111 = n. This together with the theorem above explains why cones in an (ordinary) fan have no overlap.

Example 3.6.7. Let P, VI, .•. , Vd and .1' be as in Example 3.4.11. Maximal cones are two-dimensional cones spanned by Vi-l and Vi, and we define sgnM( {i -1, i}) to be + 1 if Vi-l and Vi are in counterclockwise order and -1 otherwise. These define a multi-fan. One finds that M(P,.1') becomes a unitary toric manifold with this multi-fan. Moreover, one finds that M (P, .1') becomes an almost complex toric manifold if VI, ... , Vd are in counterclockwise order ([62]). In this case, the Todd genus of M(P, .1') agrees with the rotation number of the vectors VI, ... , Vd around the origin.

Signature

The multi-fan of M contains a lot of information on M. In fact, the theory of toric varieties says that a (ordinary) fan determines a toric variety. Namely, two toric varieties are isomorphie if their fans are same. So, the (ordinary) fan contains aIl the geometrical information on the toric variety. What can we say about unitary toric manifolds and multi-fans? Lemma 3.5.9 says that one can read off the tangential T-representations at fixed points from the multi-fan associated with a unitary toric manifold. As is well-known, it follows from the localization


theorem that aIl eharaeteristic numbers such as Todd genus and signature are determined by the tangential T-representations. Therefore, those eharaeteristic nilmbers should be deseribed in terms of multi-fan, and Theorem 3.6.6 is a formula whieh deseribes the Todd genus in terms of the multi-fan.

When dimM is divisible by 4 (Le., n is even), the signature of M is defined to be the number of positive eigenvalues minus the number of negative eigenvalues of the symmetrie bilinear form

B : Hn(M) x Hn(M) ---+ Z

defined by B(a, ß) = (aß, [MJ). Unless dimM is divisible by 4, the signature of M is defined to be zero. The signature is an oriented eobordism invariant, and the Hirzebrueh Signature Theorem teIls us how to eompute it in terms of the Pontrjagin classes of M. (See [65].)

When M is a eompaet nonsingular torie variety, it is known (see p.132 of [69]) that

n

the signature of M = ~ = :l)b4i(M) - b4i+2(M)) k=O i>O

(6.1) where ~ denotes the number of k-dimensional eones in ~ that is equal to fk-l (M) in §3.5, and bq(M) denotes the q-th Betti number of M as before. But this formula does not hold when M is a unitary torie manifold and the assoeiated multi-fan ~ has an actual overlap. It needs modifieation. See [44].

Excercise. Check the formula (6.1) for M = Cp2 (or more generaIly cpn) with the standard eomplex strueture and the T-aetion in ExampIe 3.4.7.

Low dimensional examples

Suppose the veetors VI, ... , vd in Example 3.4.11 are in comltercloekwise order. Sinee {Vi-I, vd is a Z-basis of H2(BT) for eaeh i = 1, ... , d, one has

Vi-l + Vi+1 + aivi = 0

with an integer ai. Let P and F be as in Example 3.4.11. As remarked in Example 3.6.7 we may assume that M(P,F) = M is an almost eomplex torie manifold.

3.6. Unitary toric manifolds and multi-fans

Lemma 3.6.8. Let JLi be the Poincare dual of Mi as before. Then

if li - jl = 1 or {i,j} = {l,d}

ifi = j otherwise.

113

Proof. We interpret (JLi/-Lj, [MD as the interseetion number of Mi and Mj. In the first case, Mi and Mj interseet at one point and transversally. Therefore (fLiJLj, [MD = ±1 in this case. We look at the orientation at the interseetion point. Sinee M is almost eomplex, one finds that the plus sign occurs. In the third ease, Mi and Mj do not interseet; so (/-LiJLj, [MD = 0 in this case.

The se co nd case remains. Multiply the both sides at the identity in Lemma 3.5.7 by ~ and rest riet it to H*(M). Then we have

d

0= L(U,Vi)/-LiJLj. i=1

We evaluate the both sides above on the fundamental dass [M). They reduee to

Since u is arbitrary, this implies that

Vj-1 + (JL], [M])vj + Vj+l = O.

Comparing this with the definition of aj, we have (JL], [MD = aj. 0

Theorem 3.6.9. Let M be as above.

(J) The Euler-Poincare characteristic of M is d.

(2) The signature of M is i ~ ai·

(3) The Todd genus of M is 112 (3d + 2:t=1 ad.

Proof. (1) This follows from the exeercise above Theorem 3.6.6. (2) Hirzebrueh Signature Theorem tells us that

the signature of M = ~ [MD


since M is rea14-dimensional. Here PI (M) = L1=1 fL[ by Corollary 3.5.5. Puttillg this into the above formula and using Lemma 3.6.8, the identity (2) follows.

(3) Since M is real 4-dimensional,

1 the Todd genus of M = (12 (cdM)2 + c2(M)), [MD

Here Cl (M) = L1=1 fLi and c2(M) = Li<j fLifLj by Theorem 3.6.2. Putting this into the above formula and using Lemma 3.6.8, the identity (3) follows. 0

Corollary 3.6.10.

The rotation number of VI,·.·, Vd

d

= (3d + L ad/12. i=l

Proof. As remarked before, the rotation number of VI, ... , Vd agrees with the Todd genus of M. So the corollary follows from Theorem 3.6.9 (3). 0

3.7 Moment maps and equivariant index

Moment maps

Let L be a complex T-line bundle over M. Since H:f(M) is additively generated by Us (i E AM), one can write

c[(L) = L Ci ei in H:f(M) iEA M

with integers Ci. For each i E AM we set

Fi := {u E ~ I (U,Vi) = cd·

Lemma 3.7.1. [see [44]] There is a map <I>L: MIT -+ ~ sending MdT to Fi for each i E AM, and such a map is unique up to homotopy.

3.7. Moment maps and equivariant index 115

The lemma above can be proved by a purely homotopy theoretic argument, but the map q> L arises in the context of (pre-) symplectic geometry. To elaborate it more, let PL be a principal Sl-bundle associated with L. We may think of PL as the unit circle bundle of L (for this, we need to choose aT-invariant Hermitian fiber metric on L). Since L is T-equivariant, PL has the induced T-action. Take an invariant connection I-form e on PL . Since PL has the T-action, any element v E Lie(T) defines a vector field Q on PL . The interior product iyß is a function on PL, but this function decends to a function on M. Therefore we obtain a map $L: M --+ Lie(T)* by the following formula

($ L, v) = iyß·

The map q> L is called a moment map associated with the line bundle L. However, $L is T-equivariant, the T-action on Lie(T)* being trivial; so $ L factors through M /T and pro duces the desired map q> L when Lie(T)* is naturally identified with H 2(BT; lR).

As is well-known, the exterior derivative de, which is a 2-form on PL, decends to a 2-form WL on M. The celebrated theorem by Atiyah [5] ~ Guillemin-Sternberg [38] says that the image of the moment map q>L is a convex polytope if WL is nondegenerate, i.e., if w'L is nowhere zero on M. However, the image is not necessarily convex unless WL is nondegenerate ([55], [35], [62]).

By Lemma 3.7.1, q>L(8(M/T)) is contained in the union UiEAMFi,

so we have

q> L : 8(M /T) --+ H 2 (BT; lR) \ {u} for any u 1. UFi.

Since H 2 (BT; lR) is a vector space of dimension n, H 2 (BT; lR)\{u} is homotopy equivalent to sn-I. We orient 8( M /T) and sn-l in an approapriate way and define an integer d L ( u) by

ddu):= the mapping degree of q>L : 8(M/T) --+ H 2(BT; lR)\{u}.

Geometrically speaking, dL(U) is the winding number of q>L around u. The function dL is defined on H 2 (BT; lR)\ U Fi , and takes zero on unbounded domains because 8(M/T) is compact.

Theorem 3.7.2. [[55}, !43}, !44j}

~ = the integral ofdL overH2(BT;lR). n.


When the two form WL is nondegenerate, <f>L{ß(M/T)) bounds a convex polytope in H 2(BT; lR) as remarked above. In this case, dL{u) = 1 if u lies in the interior of the convex polytope and 0 if it lies in the outside of the convex polytope, so the integral in the theorem above gives the volume of the convex polytope.

Equivariant Riemann-Roch number

The map f collapsing M to a point induces an equivariant Gysin homomorphism

f! : KT(M) ---7 KT(point) = R(T)

in equivariant K-theory, where R(T) denotes the complex representation ring of T. Then f!(L) E R(T) is called the equivariant Riemann-Roch number of Land plays an important role in topology as weIl as in algebraic geometry. See [47] and [8] for details. When L is trivial, f!(L) agrees with the Todd genus of M; so it is an integer in this case. However, f! (L) is not necessarily an integer unless L is trivial.

Denote by XU the complex one-dimensional representation corresponding to u E H 2 (BT) as in Lemma 3.3.3. Since any element of R(T) is a finite linear combination of xU's over Z, one can write

f!(L) = I: mL{u)xU

UEH2(BT)

with integers mL(u) which are zero for all but finitely many values of u. Let E : R(T) ---7 Z be a homomorphism defined by taking the sum of coefficients of XU's. It is known that

(7.1)

Our purpose is to interpret the multiplicities mL{u) in terms of the map <f> L. To do this we shift <f> L slightly so that the image of ß( M /T) misses all the lattice points in H 2 (BT). To be more specific, we set

for each i E AM. Note that no element in H 2 (BT) lies on Pi since Vi is an integral vector in H 2(BT). Similarly to Lemma 3.7.1 we have a map

~ : M/T ---7 H 2(BT; lR)

3.8. Applications to combinatorics 117

sending MdT to F! for each i, and such a map is unique up to homotopy. Thus it induces a map

~~ a(MjT) --+ H 2(BT;IR)\{u} for any u E H 2(BT).

This defines an integer, denoted hy ~ similarly to ddu).

Theorem 3.7.3. [[55], [35], [62]] mdu) = ~ /or any u E H 2(BT).

3.8 Applications to combinatorics

When M is a compact nonsingular toric variety and L is an ample T-line hundle, the image of the moment map ~ : M --+ t* = H 2 (BT; IR) is a convex polytope with vertices in the lattice H 2 (BT). So, the moment map conneets the geometry of torie varieties to eombinatories of convex polytopes. In fact , this map enables us to obtain various interesting results on eombinatories of eonvex poly tops by applying (geometrieal) results on torie varietes. Let us eonsider one speeifie example.

Pick's Formula

Let P be a polygon in IR2 with vertices in the lattiee 7L,2.

Theorem 3.8.1. [Pick's Formula] "(P) = Area(P) + ~ + 1, where "(X) denotes the number 0/ lattice points in X and Area(P) denotes the area 0/ P.

In other words, Pick's Formula teIls us that the area of P ean be computed hy eounting the numhers of lattiee points in P and in ap. The formula was found by G. Pick ahout a hundred years ago.

An interesting thing is that Pick's Formula ean be reproved when the polygon P is convex by applying results on torie varieties. The outIine of the proof is as foIlows. Let T be a two dimensional torus. We identify IR2 (resp. 7L,2) with H 2(BT; IR) (resp. H2(BT)) and view P as sitting in H 2(BT; IR). Denote the sides of P by Pi (i = 1, ... ,d). They are numbered so that the adjaeent side of Pi is Pi+l for eaeh i


in counterclockwise direction, where Pd+1 = PI. Since the vertices are lattice points, there are an element Vi E H 2(BT) and an integer Ci for each i such that

P = {u E H 2 (BT; lR) I (u, Vi) :s: Ci for i = 1, ... , d}.

For simplicity we shall assume that the set {Vi-I,vd forms a :l:-basis of H2(BT) for each i. Then the map

F: {1, ... ,d} --+ H2 (BT)

sending i to Vi is a characteristic map. Thus we obtain a toric manifold M(P, F) = M (see Section 3.4). In fact, M can be taken to be a compact nonsingular toric variety. There exists a complex T-line bundle Lover M with cf(L) = LiEAM ~ and the image of the moment

map ;r;L: M --+ H2 (BT; lR) agrees with P. Now we look at the index E(J!(L)) E z. It follows from Theorem 3.7.3 that

On the other hand, since

we have

Here the first term at the right hand side agrees with the area of P by Theorem 3.7.2, and the third term is one since M is a compact nonsingular toric variety. The anylysis of the second term needs more work, but it turns out to be ~ Putting these into (7.1), we obtain Pick's Formula.

Remark 3.8.2. Let "(PO) denote thc number of lattice points in the interior of P. Then

1 "(PO) = Area(P) - 2"(8P) + 1.


A generalized Pick 's Forrrwla

Pick's Formula and the formula in Remark 3.8.2 can be unified and generalized. We shall introduce notation to state a generalization of Pick's Formula. Let P be an oriented piecewise linear c10sed curve in ]R2 with vertices in 71..2 and with sign assigned to each side Pi of P (i =

1, ... , d). Each side is numbered so that the next side of Pi is Pi+ 1

for each i. (Here next means when we move on P according to the orient at ion on P.) The sign assigned to Pi is denoted by sgn(Pi). Let ni (i = 1, ... , d) be anormal vector to Pi such that the 90 degree (counterc1ockwise) rotation of sgn(Pdni has the same direction as Pi. The winding number of P around a point in ]R2\p defines a locally constant function dp on ]R2\p. Since P is compact, dp takes zero on unbounded domains in ]R2. We consider three invariants of P:

A(P) : = the integral of dp over ]R2,

d

B(P) : = L ~ - 1), i=l

C(P) : = the rotation number of the sequence of normal vectors

We say that P is simple if P has no self-intersection, sgn(Pd is positive for any i , and P is oriented so that the domain bounded by P lies on the left-hand side of P when we go round in the direction of the orientation of P. If P is simple, then A(P) is the area of the domain bounded by P, B(P) is the number of lattice points in P and C(P) = l.

We now define an integer U(P) which coincides with the number of lattice points in the domain bounded by P when P is simple. Let P' be an oriented piecewise linear c10sed curve in ]R2 obtained from P by translating each side Pi slightly in the direction of ni. It miss es lattice points, so that the winding number dp , (u) is defined for any lattice point u. It vanishes for all but finitely many lattice points and depends only on P. We define

U(P) := L dp,(u).

uE7I..2

Theorem 3.8.3. (A generalized Pick 's Formula ([62]))

U(P) = A(P) + ~ + C(P).


Excercise. Check the following.

(1) The formula above reduces to Pick's Formula when P is simple.

(2) Let po be P with reversed signs on all sides of P. Then

~ = A(P) - ~ + C(P).

(3) When P is simple, the above formula (2) reduces to the formula in Ftemark 3.8.2.

Excercise. Check the generalized Pick's formula for the following star shaped polygon with self-intersections by taking arbitrary signs on sides.

Ehrhart polynomial

A generalization of Pick's Formula to a higher dimensional case was studied hy Ehrhart. Let P be an n-dimensional (not necessarily simple) lattice convex polytope in IRn . For a positive integer q, we set

qP := {qu I u E P}

which is again a convex polytope. When n = 2, Pick's Formula implies that

1 ~ = Area(p)q2 + ~ + l. (8.1)

This is generalized by Ehrhart to

Theorem 3.8.4. [Ehrhart} ~ is a polynomial in q of degree n with the volume of P as the coefficient of qn and 1 as the constant term.


Sinee ~ is a polynomial in q, one ean plug -q in the polynomial. Then it gives ~ i.e., (_l)n times the number oflattiee points in the interior of qP. (Cf. Remark 3.8.2 and (8.1).) This fact is known as the inversion formula. See [24], [69] or [32] for more details.

The Ehrhart's theorem above ean be proved by a purely eombinatorial method, but it ean also be reproved using results on torie varieties as is shown in the proof of Pick's Formula. One ean expeet a generalization of the Ehrhart's theorem to more general polytopes like we had the generalized Piek's Formula. This is done in [44] to what we eall multi-polytopes (which we do not define here).

The theory developed so far has more applieations to eombinatories like the theory of torie varieties had. We refer the reader to [44], [63], [64].

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Index

Almost effective, 3

Almost complex toric manifold, 108

Alomst free, 3

Borel cOllstruction, 2, 6 Borel formula, 26

c-Hamiltonian, 36 c-Kähler, 36 c-symplectic, 35 Cartan model, 31 Characteristic map, 100 Characteristic submanifolds, 99 Cone

dual, 44 nonsingular , 54 scrap,44 simplicial, 54

Effective, 3 Ehrhart polYllomial, 120 Equivariant

characteristic classes, 91 cohomology, 88 GYSill homomorphism, 20, 21 Riemann-Roch number, 116

Face, 45 Face ring or Stanley-Reisner ring,

103

Facet, 97 Fan, 47

nonsingular , 54 simplicial, 54

Fixed point set, 3 Frankel's Theorem, 33 Free, 3 Fundamental group, 66

Generalized Pick's formula, 119 Group action, 2

Hamiltonian, 30 Hamiltonian group action, 31 Hamiltonian vector field, 29 Hirzebruch surface, 64, 99

Index of a Morse-Bott function, 32

Ineffective kernei, 3 Invariant form, 31 Isotropy subgroup, 2

Jones-Rawnsley Theorem, 34

Lefschetz condition, 36 Line bundle, 70

ample, 74 very ample, 74

Localization Theorem, 13, 90 Locally Hamiltonian vector field,

29 Locally toric manifold, 96

130

Manifold with corners, 97 Moment map, 31, 84, 114, 115 Morse-Bott function, 32 Multi-fans, 109, 110

Orbifold, 55 Orbit, 2

map, 3 space,3

Picard group, 69 Pick's formula, 85, 117, 118 Poincare duality, 20 Poisson bracket, 29

Quasi toric manifold, 87

Semi-free, 3 Signature, 111 Simple convex polytope, 97 Symplectic

group action, 31 form, 28 gradient, 29 manifold, 28

Tangent bundle, 79 TNHZ, 7 Todd dass, 79 Todd genus, 80, 108 Topological splitting principle, 26 Toric manifold, 97 Toric variety, 85

affine, 45 complete, 56 normal, 56 smooth,54

Torus manifold, 86

Uniform torus action, 40 Unitary toric manifold, 107 Unitary torus manifold, 108

INDEX

Transformation Groups: Symplectic Torus Actions and Toric Manifolds

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