As is well known, the correspondence (2) above is generalized to compact symplectic manifolds with torus actions admitting moment maps. A symplectic m

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1 TORUS ACTIONS ON MANIFOLDS Mikiya Masuda An overview. In this series of lectures I intend to develope the theory of toric varieties, which is a bridge between algebraic geometry and combinatorics, from a topological point of view. In short, our aim is to construct a bridge between topology and combinatorics. A toric variety of dimension n is a normal complex algebraic variety with an action of (C ) n having a dense orbit, where C = C nf0g. Ane space C n and a complex projective space C P n with standard (C ) n -actions are typical examples of toric varieties, and other toric varieties may be viewed as a generalization of these examples. Although the condition that the action of (C ) n has a dense orbit is very restrictive, toric varieties are abundant. The basic theory of toric varieties was established around early 70's by Demazure, Mumford etc., and Miyake-Oda, see [Fu], [O]. It says that (1) there is a one-to-one correspondence between toric varieties and combinatorial objects called fans, (2) a compact nonsingular toric variety together with an ample complex line bundle corresponds to a convex polytope through a map called the moment map. After the basic theory was established, many interesting applications to combinatorics (especially to convex polytopes) have been found. A simple but intriguing application is Pick's Formula. It says that if P is a convex polytope in R 2 with vertices in Z 2, then Area(P ) = ](P ) ; 1 2 ](@P) ; 1 where ](X) denote the number of lattice points in X. Pick's Formula can be proved by an elementary method, but it can also be proved using the correspondence (2) above. However, Pick's Formula holds even if P is not convex and this extended case is not covered by the theory of toric varieties. This awkwardness suggests the existence of a theory which allows us to prove Pick's Formula in full generality, and we will see that such a theory does exist in topology. To be more precise, our geometrical object is a toric manifold which is a smooth manifold of dimension 2n with an eective action of a compact torus T = (S 1 ) n having a convex polytope as the orbit space. A compact nonsingular toric variety with the restricted T -action provides an example of a toric manifold. The tools we use are purely topological. Because of this, our argument works for a larger class than compact nonsingular toric varieties. Interestingly, new (or more general) combinatorial objects appear in this extended context, i.e., fans are replaced by multi-fans in (1) above and convex polytopes are replaced by twisted polytopes in (2) above. 1 Typeset by AMS-TEX

2 As is well known, the correspondence (2) above is generalized to compact symplectic manifolds with torus actions admitting moment maps. A symplectic manifold M is a smooth manifold of even dimension, say 2n, with a nondegenerate closed two form!. Here! is said to be nondegenerate if the 2n-form! n is nowhere zero on M. A compact nonsingular toric variety together with an ample line bundle L provides an example of a symplectic manifold with an action of T admitting a moment map, the two form being the rst Chern form of L. In our case, the complex line bundle L is arbitrary, and the rst Chern form of L is often degenerate although it is closed. Therefore, our toric manifold together with a complex line bundle does not necessarily provide an example of a symplectic manifold. 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 variety. There are two defects in our theory for the moment. The rst 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 V- manifold). Orbifolds are close to smooth manifolds, and many results on smooth manifolds would hold for orbifolds with a little modication. However, it would be hard to treat toric varieties with tough singularities from a topological point of view. The second defect is that the correspondence (2) above is not one-to-one in our case. We can associate a multi-fan with a unitray toric manifold (see x7), but the correspondence is not injective. Namely, two unitary toric manifolds with the same multi-fan are not necessarily isomorphic (although they are equivariantly unitary cobordant). Furthermore, we do not know what types of multi-fans arise from unitary toric manifolds. It is a fundamental problem in our theory to characterize multi-fans obtained from unitary toric manifolds. 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 [Bri]) and the theory is developing, and there is a certain convexity theorem for actions of compact non-abelian groups on symplectic manifolds (see [Ki2]). But it seems to me that the non-abelian analogue of the theory of toric varieties is not yet fully exploited. x1. Basic notation and terminology. In this series of lectures we will work in the smooth(=c 1 ) category, so the word \smooth" will often be omitted. We begin with the review of basic notation and terminology on Lie group actions on smooth manifolds. Lie groups 2

3 Let G be a Lie group, that is, G is a manifold, and a group, and the map G G! G sending (a b)! ab ;1 is smooth. It is known that a closed subgroup of a Lie group G is a Lie group, and if H is a closed normal subgroup of G, then the quotient group G=H is also a Lie group. Two Lie groups are isomorphic if there is a map between them which is a dieomorphism and a group isomorphism. Example 1.1. The Euclidean space R n is a Lie group, the group operation being the usual addition. The lattice Z n of R n is a closed subgroup of R n and the quotient group R n =Z n is a compact Lie group called an n-dimensional torus. Excercise 1.2. The group of complex numbers with unit length, that is S 1 as a manifold, is a Lie group under complex multiplication, and their n-fold cartesian product (S 1 ) n is also a Lie group. Show that R n =Z n is isomorphic to (S 1 ) n as Lie groups. There are many Lie groups obtained from matrices. For instance, the group of all real or complex matrices (of a size n) with nonzero determinants is a (noncompact) Lie group. They are called a general linear group and denoted by GL(n R ) or GL(n C ) respectively. The subgroup O(n) of GL(n R ) consisting of all orthogonal matrices is a compact Lie group. Any closed subgroup of O(n) is also a compact Lie group. On the other hand, any compact Lie group is known to be a closed subgroup of O(n) for some n, which is a consequence of the Peter-Weyl theorem (see p.5 in [Hs]). Denition of a group action Let M be a manifold. An action of G on M is a smooth map ' : G M! M satisfying these two conditions: (A1) '(g 1 '(g 2 x)) = '(g 1 g 2 x) (A2) '(e x) = x for any g 1 g 2 2 G and any x 2 M, where e denotes the identity element of G. A manifold with a G-action is called a G-manifold. Excercise 1.3. Let Di(M) be the group of all dieomorphisms of M. Prove that (1) '(g ) 2 Di(M) for each g 2 G. (2) The conditions (A1) and (A2) above are equivalent to saying that the map ~' : G! Di(M) sending g to '(g ) is a homomorphism. Example 1.4. Let V be a vector space of nite dimension over a eld K (e.g. K = R or C ), and let GL(V ) be the group of all K -linear automorphisms of V. Let be a homomorphism : G! GL(V ) (i.e., a representation of G). Then the map ' : G V! V dened by '(g x) = (g)(x) is an action of G on V. This action is called a linear action of G on V. The vector space V with a linear G-action is sometimes specied as a G-representation space (or a G-module), but we often omit the word \space", i.e., a G-representation space is also called a G-representation. Basic terminology and notation We often do not specify the action map '. For instance, we abbreviate '(g x) as gx, and when ' is understood, we say that G acts on M. A map f : M! M 0 3

4 between two G-manifolds is said to be (G-)equivariant if f(gx) = gf(x) for any g 2 G and any x 2 M. A G-equivariant dieomorphism is called a G-dieomorphism. For each x 2 M we set G x := fg 2 G j gx = xg and call it the isotropy subgroup (or stabilizer) of G at x. The G-action on M is said to be trivial if G x = G for all x 2 M. The opposite case is when G x = feg for all x 2 M. In this case the action is said to be free. Excercise 1.5. For an n-tuple (m 1 : : : m n ) 2 Z n, dene an action of g 2 S 1 on S 2n;1 = f(z 1 : : : z k ) 2 C k j jz 1 j jz k j 2 = 1g by (z 1 : : : z k )! (g m 1 z 1 : : : g m n z n ): Find n-tuples (m 1 : : : m n ) when the action is free. The G-action on M is said to be eective if \ x2m G x = feg, in other words, if the homomorphism ~' : G! Di(M) is injective. A representation : G! GL(V ) is called faithful if is injective, in other words, if the associated linear action of G on V is eective. Excercise 1.6. Prove that (1) The intersection \ x2m G x is a closed normal subgroup of G. (2) g 2 \ x2m G x if and only if '(g ) = the identity in Di(M). Remark 1.7. Since the action of \ x2m G x is trivial, the action of G reduces to that of a quotient group G= \ x2m G x, which is eective. Therefore we may assume that the G-action on M is eective from the beginning without loss of generality. For a subgroup H of G, we set and call it the H-xed point set of M. We set M H := fx 2 M j G x Hg Gx := fgx j g 2 Gg M and call it the orbit of x. The set of all orbits in M is denoted by M=G, where an orbit is viewed as a point in M=G. There is a natural map from M onto M=G and we give M=G the quotient topology. The topological space M=G is called the orbit space of the G-manifold M. The orbit space reects the property of the G-action to some extent. It is a rather complicated topological space but happens to be a manifold (possibly with boundary). Example 1.8. Let S 1 act on S 2 as rotations xing the north and south poles. The orbit space S 2 =S 1 is homeomorphic to a closed interval [0 1]. Excercise 1.9. Consider a linear action of S 1 on C n dened by (z 1 : : : z n )! (gz 1 : : : gz n ): Prove that the orbit space C n =S 1 is not homeomorphic to a manifold (with boundary) if n 3. What happens when n = 1 or 2? 4

5 x2. Tangential representations and exponential maps. Tangential representations An advantage to work in the smooth category is the existence of the tangent space x M at each point x 2 M. The collection of the tangent spaces x M over x 2 M forms the tangent bundle M of M. The tangent space x M describes the G-action on a neighborhood of x in M. Let me explain this in more details. The dierential of g 2 G denes a linear map: x M! gx M, so that x M is a (real) representation of the isotropy subgroup G x. This representation is called the tangential representation of G x at x. Invariant Riemannian metric We assume that G is compact, which is essential in the arguments developed in this section. It is used for Lemma 2.1. There is a G-invariant Riemannian metric on M. Idea of proof. Recall that a Riemannian metric on M is a choice of an inner product on each tangent space x M, which vary smoothly on x. Every manifold admits a Riemannian metric. Take a Riemannian metric h i on M arbitrary and average it on G using the Haar measure dg which is invariant under right translations on G. Namely we dene a new Riemannian metric ( ) on M by (u v) := Z G hgu gvi dg for u v 2 x M. One checks that ( ) is G-invariant. The compactness assumption on G is necessary for the integral to be dened. Remark 2.2. When G is nite, the integral above reduces to 1 jgj X g2g hgu gvi for u v 2 x M. Exponential map A Riemannian metric on M enables us to measure the lengths of curves on M. The geodesics are curves that locally minimize length. Using geodesics, the exponential map exp x : x M! M is dened as follows. For v 2 x M, let (t) be the unique geodesic in M with the properties (0) = x and d dt (0) = v. We then dene exp x(v) := (1). The exponential map exp x is a dieomorphism from a neighborhood of the origin in x M onto a neighborhood of x in M. Excercise 2.3. Prove that if the chosen Riemannian metric is G-invariant, then the associated map exp x is G x -equivariant. 5

6 Lemma 2.4. For a subgroup H of G, M H is a submanifold of G. Idea of proof. Let x 2 M H. Then G x H. Choose a G-invariant Riemannian metric on M. Then the associated map exp x : x M! M is G x -equivariant (in particular, H-equivariant as G x H), so exp x restricts to a map : ( x M) H! M H, which is a homeomorphism from a neighborhood of the origin in ( x M) H onto a neighborhood of x in M H. This gives a local coordinate of M H since ( x M) H is a vector space. Lemma 2.5. Let H be a closed subgroup of G. If x and y lie in a same connected component of M H, then x M and y M are isomorphic as H-representations. Idea of proof. Since x and y lie in a same connected component of M H, there is a continuous path c(t) (0 t 1) in M H connecting x and y. Therefore we obtain a family of H-representations c(t) M which connect x M and y M and vary continuously on c(t). On the other hand, it is known that the isomorphism classes of H-representations are discrete since H is compact. This means that H- representations in a same connected component are mutually isomorphic. Therefore x M = y M as H-representations. Lemma 2.6. Suppose that M is connected. Then the tangential G x -representation x M is faithful at any x 2 M if (and only if) the G-action on M is eective. Proof. Let H be a subgroup of G x which acts on x M trivially. Since the exponential map exp x : x M! M is a G x -equivariant dieomorphism from a neighborhood of the origin in x M onto a neighborhood of x in M, H xes a neighborhood of x in M pointwise. On the other hand, M H is a submanifold of M by Lemma 2.4. Since M is connected, one concludes that M H = M hence H = feg because the G-action on M is eective. This proves that the tangential G x -representation x M is faithful. Slice Theorem The subspace x (Gx) of x M tangent to the orbit Gx is G x -invariant. Choose a G x -invariant inner product on x M and let W be a G x -subrepresentation of x M which is orthogonal to x (Gx). The isomorphism class of W is independent of the choice of the G x -invariant inner product on x M. In fact, W is isomorphic to a quotient representation x M= x (Gx). We consider an orbit space (G W )=G x of the G x -action dened by (g w)! (gh ;1 hw) for h 2 G x. The orbit space is often expressed as G Gx W and sometimes called the balanced product of G x -spaces G and W. It has a G-action induced by (g w)! (g 0 g w) for g 0 2 G. Theorem 2.7 (Slice Theorem). There exists a G-equivariant dieomorphism from G Gx W onto a neighborhood of the orbit Gx in M which maps (g 0) to gx. Corollary 2.8. The orbit space W=G x is homeomorphic to a neighborhood of x in M=G. 6

7 x3. Equivariant cohomology. In this section G is a compact Lie group and M is a compact G-manifold although the argument works in a more general setting. Universal principal G-bundles Let EG be a contractible topological space with a free G-action. We understand that G is acting on EG from the right. Let BG be its orbit space. Then EG! BG is a principal G-bundle and any principal G-bundle over a topological space X is obtained by pulling it back by a map from X to BG, so EG! BG is called the universal principal G-bundle and BG the classifying space of principal G-bundles. Excercise 3.1. Let G i (i = 1 : : : n) be compact Lie groups and let EG i! BG i be the universal principal Q Q Q n G i -bundle for each i. Check that EG n i=1 i! BG i=1 i n is the universal principal i=1 G i-bundle. The orbit space S 2n;1 =S 1 of the free S 1 -action on S 2n;1 in Excercise 1.5 (with m 1 = = m n = 1) is a complex projective space C P n;1. There are natural inclusions S 1 S 3 S 5 : : : C P 0 C P 1 C P 2 : : : and we consider their inductive limits S 1 = [ 1 n=1 S2n;1 and C P 1 = [ 1 n=1c P n;1. (We give them the weak topology, so that they become topological spaces.) One checks that S 1! C P 1 is a universal principal S 1 -bundle. Therefore their n-fold cartesian product gives a universal principal (S 1 ) n -bundle. Remark 3.2. Note that S 3 = S 1 S 1, S 5 = S 1 S 1 S 1 and so on, where denotes the join. Therefore S 1 = S 1 S 1. More generally, it is known that EG = G G : : : for any topological group G. Equivariant cohomology Throughout this note, homology and cohomology groups are taken with Z coef- cients unless otherwise stated. The balanced product EG G M (here, the action of g 2 G on (z x) 2 EG M is given by (zg ;1 gx)) is denoted by M G. We set H q G (M) := Hq (M G ) and call it the (q-th) equivariant cohomology group of the G-manifold M. The equivariant cohomology was introduced by A. Borel around 60's ([Bo]). He used it to reprove the famous Smith xed point theorem. Around 70's it was extensively used by Bredon (see [Bre, Chapter VII]), Hsiang brothers (see [Hs]) and others to study actions on appropriate topological spaces. In 80's it was realized by Atiyah, Bott [AB] and Kirwan [Ki1] that the equivariant cohomology ts well to the study of actions on symplectic manifolds. As is well known, H q (M R ) is isomorphic to the de Rham cohomology of M dened using forms on M. Although M G is not a manifold, there is a de Rham cohomology version of equivariant cohomology, see [AB]. Excercise 3.3. Suppose that the G-action on M is trivial. Then prove that (1) M G is homeomorphic to BG M. (2) H G (M) = H (BG) H (M) if H (BG) has no torsion. 7

8 The projection from M G = EG G M onto the rst factor EG=G = BG denes a bration with ber M: M! M G ;! BG One can use the Serre spectralsequence of this bration to compute the cohomology of the total space, that is HG (M). Generally speaking, it is not so easy to compute (M) but there is a very simple situation. H G Lemma 3.4. If the cohomology groups of M and BG are torsion free and vanish in odd degrees, then H G (M) = H (BG) H (M). The isomorphism in the lemma above is an isomorphism as H (BG)-modules, where HG (M) is viewed as an H (BG)-module through : H (BG)! HG (M). However HG (M) has a ner structure. It is a ring under cup product, furthermore the ring structure together with the H (BG)-module structure makes HG (M) as an algebra over H (BG). This algebra structure cannot be read o from the spectral sequence argument and contains more information than the module structure. As we will see later, the H (BG)-algebra structure on HG (M) reects the G-action on M pretty well, and we need to know the algebra structure to determine the ordinary cohomology group H (M) through the restriction map: HG (M)! H (M). Localization Theorem A benet of equivariant cohomology is that a global invariant lying in the equivariant cohomology HG (M) can often be determined by local data around the xed point set M G by virtue of the following localization theorem (see [Hs, p.40]). Theorem 3.5 (Localization Theorem). Let S = H (BG)nf0g. Then the localized restriction homomorphism S ;1 H G(M)! S ;1 H G(M G ) is an isomorphism, where S ;1 N for an S-module N consists of all fractions fm=s j m 2 N s 2 Sg with the usual identication m 1 =s 1 = m 2 =s 2 if and only if ss 1 m 2 = ss 2 m 1 for some s 2 S. Suppose that the ring H (BG) is an integral domain (i.e., there is no zero divisor except zero). Then H G (M G ) = H (BG) H (M G ) has no S-torsion. This together with Localization Theorem implies that the kernel of the restriction map: H G (M)! H G (M G ) consists of all S-torsion elements in H G (M). Corollary 3.6. Suppose that H (BG) is an integral domain with H odd (BG) = 0, H (M) is torision free, and H odd (M) = 0. Then the restriction map: H G (M)! H G (M G ) is injective. Proof. The assumption together with Lemma 3.4 implies that both HG (M) and HG (M G ) have no S-torsion. Therefore the corollary follows from Localization Theorem. Equivariant characterictic classes 8

9 There are three characteristic classes for vector bundles. They are Euler class, Chern class and Pontrjagin class (see [MS]). Let us review them briey. Let : E! M be a vector bundle. (1) The Euler class e(e) is dened for an oriented vector bundle E, and lies in H q (M) where q is the ber dimension of E. If the orientation on E is reversed, then the Euler class changes the sign. (2) The (total) Chern class c(e) = 1+c 1 (E) + c 2 (E) + : : : is dened for a complex vector bundle E, and the k-th Chern class c k (E) lies in H 2k (M). If r is the complex ber dimension of E, then c k (E) = 0 for k > r and the top Chern class c r (E) agrees with the Euler class of the underlying oriented real vector bundle of E. (3) The (total) Pontrjagin class p(e) = 1 + p 1 (E) + p 2 (E) + : : : is dened for a real vector bundle E, and the k-th Pontrjagin class p k (E) lies in H 4k (M). In fact, p k (E) is dened to be (;1) k c 2k (E C ). Using these characteristic classes, one can dene equivariant characteristic classes like we dened equivariant cohomology using ordinary cohomology. Before we give the denition, we shall review the denition of a G-vector bundle. Denition 3.7. A vector bundle : E! M is a G-vector bundle if (1) G acts on both E and M, and the map is equivariant, (2) each element g 2 G maps ;1 (x) to ;1 (gx) linearly for any x 2 M. Remark 3.8. If E admits an additional structure such as orientation or complex structure, then we require in (2) above that g : ;1 (x)! ;1 (gx) preserves the structure. Suppose that : E! M is a G-vector bundle. Then it produces a vector bundle G = 1 G : E G = EG G E! M G = EG G M of the same ber dimension as E. Excercise 3.9. Choose a point in E and dene a bundle isomorphism from E to E G. The additional structure on E such as orientation and complex structure is inherited to E G. The equivariant Euler class of an oriented G-vector bundle E, denoted e G (E), is dened to be the ordinary Euler class of E G : e G (E) := e(e G ) 2 H q G (M) where q is the ber dimension of E. A G-representation V is a G-vector bundle over a point, so e G (V ) lies in H q (BG) where q = dim V. The equivariant Chern (resp. Pontrjagin) class of a complex (resp. real) G-vector bundle will be dened in a similar fashion : c G (E) := c(e G ) and p G (E) := p(e G ): When E is the tangent bundle M of M, then e G (M), c G (M) and p G (M) are often denoted by e G (M), c G (M) and p G (M) respectively. Equivariant Gysin homomorphism 9

10 Let X and Y be closed oriented G-manifolds. For any equivariant map f : X! Y, the equivariant Gysin homomorphism f! : H q G (X)! Hq+r G (Y ) is dened, where r = dim X ; dim Y, and has the same properties as the ordinary Gysin homomorphism (see [Ka]). For the reader's convenience, I list the useful properties in the following. Property 1. Let h : Y! Z be an equivariant map between closed oriented G- manifolds. Then (h f)! = h! f!. Property 2. f! (uf (v)) = f! (u)v for u 2 HG (X) and v 2 H G (Y ). Property 3. Let X 1 and X 2 be closed oriented G-submanifolds of X which intersect transversally. Consider the following diagram of inclusion maps: i X 1 \ X 0 2 ;;;;! X 1 j? y? y X 2 j 0 ;;;;! X: Then we have an identity i 0! j = i j 0!. In particular, if X 1 and X 2 have no intersection, then i j 0! = 0. Property 4. If X is a closed oriented G-submanifold of Y and f is the inclusion, then f f! (1) = e G () where 1 2 HG 0 (X) is the identity and denotes the normal G-vector bundle of X in Y. i x4. Representations of a torus. Complex representations of a torus Henceforth we shall denote the n-dimensional torus (S 1 ) n by T. A complex one-dimensional representation of T is a homomorphism from T to GL(1 C ) = C. Since T is compact, the image of the homomorphism lies in the maximal compact subgroup of C, that is S 1. Denote by Hom(T S 1 ) the set of homomorphisms from T to S 1, in other words, the set of complex one-dimensional representations. It forms an abelian group under the multiplication on the target space S 1. 10

11 Lemma 4.1. Any complex one-dimensional representation of T = (S 1 ) n is given by (t 1 : : : t n ) 2 T! ny k=1 t m k k 2 S 1 with some n-tuple (m 1 : : : m n ) 2 Z n. Moreover, dierent n-tuples produce nonisomorphic complex one-dimensional T -representations. In particular, Hom(T S 1 ) is isomorphic to Z n. We shall interpret Hom(T S 1 ) in terms of topology. For f 2 Hom(T S 1 ) let V f be the associated complex one-dimensional T -representation space. Since c T 1 (V f ) = e T (V f ) is an element of H 2 (BT), we obtain a map One checks that is an isomorphism. : Hom(T S 1 )! H 2 (BT): Since H (C P 1 ) is a polynomial ring over Z in a generator of H 2 (C P 1 ), it follows from Kunneth formula that H (BT) is a polynomial ring over Z in n elements of H 2 (BT). In particualr, H 2 (BT) is isomorphic to Z n, so gives another identication of Hom(T S 1 ) with Z n. 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. Real representations of a torus When we forget the complex structure on a complex representation, we obtain a real representation. This is called the realication of a complex representation. In the above, we have observed all complex one-dimensional representations of T. They produce real two-dimensional representations of T. But dierent complex one-dimensional representations of T may produce isomorphic real representations of T. Excercise 4.2. For an n-tuple m = (m 1 : : : m n ) 2 Z n, denote by m the complex one-dimensional T -representation dened in Lemma 4.1. Then m is isomorphic to m 0 as real T -representations if and only if m 0 = m. The representation theory says (and it is not dicult to see) that any irreducible nontrivial real T -representation is two-dimensional and obtained as realication of a complex one-dimensional representation. Through the isomorphism, this together with the excercise above implies Lemma 4.3. The set of isomorphism classes of irreducible real T -representations bijectively corresponds to a quotient set H 2 (BT)= s where s if and only if =. Circle subgroups of a torus 11

12 We have observed that Hom(T S 1 ) = H 2 (BT). A dual version of this isomorphism is that _ : Hom(S 1 T ) = H 2 (BT): In fact, the map _ is dened as follows. Consider the pairing Hom(T S 1 ) Hom(S 1 T )! Hom(S 1 S 1 ) = Z dened by composition of maps. It is nondegenerate, so we have isomorphisms Hom(S 1 T ) = Hom(Hom(T S 1 ) Z) = Hom(H 2 (BT) Z) = H 2 (BT): The composition of these isomorphisms is our _. The following would be clear from this denition. Lemma 4.4. For u 2 H 2 (BT) and v 2 H 2 (BT), let u 2 Hom(T S 1 ) and v 2 Hom(S 1 T ) be the corresponding elements via and _ respectively. Then ( u v )(z) = z hu vi for z 2 S 1. The image of a nontrivial homomorphism from S 1 to T is a circle subgroup of T, but two such homomorphisms may determine the same circle subgroup of T. Through _, one sees that Lemma 4.5. Two nonzero elements 1 2 in H 2 (BT) determine the same circle subgroup of T if and only if they span the same \line" through the origin in H 2 (BT), in other words, if and only if m 1 1 = m 2 2 with some nonzero integers m 1 m 2. Orbit spaces of T -representations By the slice theorem the orbit space of a G-manifold is locally homeomorphic to that of a representation. Later we will be concerned with the case where G = T, so we shall study the orbit space of a T -representation. Lemma 4.6. Let V be a faithful real T -representation. Then dim V T dim V ; 2 dim T, in particular, 2 dim T dim V. Proof. Take a T -invariant inner product on V. Then V decomposes into the direct sum of V T and its orthogonal complement (V T )?. Since a nontrivial irreducible real T -representation is two-dimensional and obtained as the realication of a complex one-dimensional representation, one may think of (V T )? as the direct sum of complex one-dimensional T -representations. Let 1 : : : k be their characters, where k = dim C (V T )? Q. The faithfulness of the T -representation V is equivalent k to the homomorphism i=1 i : T! (S 1 ) k being injective. Therefore dim T k. Since dim V = dim V T + 2k, the lemma follows. The orbit space of a T -representation V is not necessarily a manifold (with boundary), but it is, in the following extreme case. 12

13 Lemma 4.7. Suppose that V is a faithful real T -representation and 2 dim T = dim V. Then the orbit space V=T is homeomorphic to (R + ) n where R + denotes the nonnegative real numbers. Proof. Since 2 dim T = dim V, V T = f0g by Lemma 4.6 and V may be viewed as the direct sum of n complex Q one-dimensional T -representations with characters n i 's. The homomorphism i=1 i : T! (S 1 ) n is injective as remarked in the proof of the above lemma, but since T and (S 1 ) n have the same dimension, the homomorrphism must be an isomorphism. This implies that the linear T -action on V is isomorphic to the standard linear action of (S 1 ) n on C n dened by (z 1 : : : z n )! (g 1 z 1 : : : g n z n ) where (z 1 : : : z n ) 2 C n and (g 1 : : : g n ) 2 (S 1 ) n. The orbit space of the latter is easily seen to be (R + ) n. In fact, a map from C n to (R + ) n sending (z 1 : : : z n ) to (jz 1 j : : : jz n j) induces a homeomorphism from C n =T to (R + ) n. x5. Toric manifolds. Let T be an n-dimensional torus as before and let M be a T -manifold. Henceforth the T -action on M will be assumed to be eective although we do not mention it. We begin with Lemma 5.1. If M T 6=, then dim M T dim M;2 dim T, in particular, 2 dim T dim M, where dim M T denotes the maximum of the dimensions of connected components in M T. Proof. Let x be a point of a connected component in M T of the maximum dimension. Since the action is eective, the tangential representation x M is faithful so dim x M T dim x M ; 2 dim T by Lemma 4.6. Since dim x M T = dim M T and dim x M = dim M, the lemma follows. Excercise 5.2. Give an example of a T -manifold M such that the conclusion of the above lemma does not hold if the assumption M T 6= is dropped. Locally toric manifolds Denition 5.3. A T -manifold M of dimension 2n is called a locally toric manifold if the following two conditions are satised: (1) dim T = n (hence M T consists of isolated points by Lemma 5.1 if it is nonempty). (2) At each point x 2 M, the isotropy subgroup T x is a subtorus of T and dim( x M) T x = 2(n ; dim T x ). The condition (2) above is rather technical, but it ensures that the orbit space M=T is nice. The following theorem follows from Slice Theorem and Lemma

14 Lemma 5.4. If M is a locally toric manifold of dimension 2n, then the orbit space M=T is an n-dimensional manifold with corners. Here a topological space X is said to be a manifold with corners of dimension n if it is given a set of pairs f(u )g which satises the following conditions: (1) fu g is an open covering of X, (2) : U! (R + ) n is a homeomorphism onto the image, (3) if U \ U 6=, then ;1 : (U \ U )! (U \ U ) extends to a smooth map on a neighborhood of (U \ U ) in R n. The local coordinate map in (2) above is not required to be surjective. Since (R + ) n contains an open set dieomorphic to R n, a usual 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. Toric manifolds An important example of a manifold with corners is a simple convex polytope. A convex polytope P is the convex hull of a nite set of points in R n. We may assume that the points are in a general position (i.e., they are not in an ane hyperplane) because if they are, then we may think of them as points in R n;1. A (proper) face of P is the intersection with an ane hyperplane in R n such that P sits in a region divided by the hyperplane. A face of dimension n ; 1 is called a facet of P. We say that P is simple if each vertex in P lies on exactly n facets. Excercise 5.5. There are ve regular convex polytopes in R 3 (tetrahedron, hexahedron, octahedron, dodecahedron, icosahedron). (1) Which one is a simple convex polytope? (2) Find a simple convex polytope of dimension three which is not a regular convex polytope. (If you cannot nd it, look at a soccer ball.) Denition 5.6. Following Davis-Januszkiewicz [DJ], we call a locally toric manifold M a toric manifold if the orbit space M=T is dieomorphic to a simple convex polytope. Remark 5.7. The condition that M=T is dieomorphic to a simple convex polytope is rather strong. Many arguments developed below work in a more general setting, see [HM]. Example 5.8. The complex projective space C P n with the action of T = (S 1 ) n dened by [z 1 : : : z n+1 ]! [g 1 z 1 : : : g n z n z n+1 ] where [z 1 : : : z n+1 ] 2 C P n is the homogeneous coordinate, is a toric manifold. The map from C P n to R n+1 dened by [z 1 : : : z n+1 ]! 1 P n+1 i=1 jz ij 2 (jz 1j 2 : : : jz n+1 j 2 ) induces a dieomorphism from C P n =T to the standard n-simplex. 14

15 Example 5.9. Let S 1 act on S 2 as rotations xing the north and south poles. Then the n-fold cartesian product (S 2 ) n has an action of T = (S 1 ) n. One easily checks that this is a toric manifold. The orbit space is dieomorphic to [0 1] n and the T -xed point set consists of 2 n points. Excercise Consider S 2n = f(z 1 : : : z n x) 2 C n R j with the action of T = (S 1 ) n dened by nx i=1 (z : : : z n x)! (g 1 z 1 : : : g n z n x): jz i j 2 + x 2 = 1g Show that S 2n with this T -action is a locally toric manifold but not a toric manifold if n 2. A toric manifold is a topological counterpart to a compact nonsingular toric variety in algebraic geometry. A compact nonsingular toric variety with the restricted T -action provides an example of a toric manifold. But there are many toric manifolds which do not arise from nonsingular toric varieties as we will see later. Let M be a toric manifold. Characteristic submanifolds Denition A connected codimension two submanifold of M is called characteristic if it is left xed pointwise under a certain circle subgroup of T. Let M be an index set parametrizing characteristic submanifolds of M, i.e., a characteristic submanifold of M will be denoted by M i (i 2 M ). For a subset I of M the intersection \ i2i M i will be abbreviated as M I. Excercise Prove that (1) The T -action is free on Mn [ i2m M i. (2) If M I 6=, then M i 's (i 2 I) intersect transversely. (Hence dim M I = 2(n ; jij) and jij n, where jij denotes the cardinality of I.) We will denote the circle subgroup which xes M i pointwise by T i, and the subtorus generated by T i (i 2 M ) by T I. Note that dim T I = jij and T I xes M I pointwise. Example Consider C P n with the T -action in Example 5.8. One checks that the submanifold (C P n ) i dened by z i = 0 is characteristic, so CP n = f1 : : : n + 1g. The circle subgroup T i which xes (C P n ) i pointwise is the subgroup f(1 : : : g i : : : 1) j g i 2 S 1 g for i n and the diagonal circle subgroup f(g : : : g) j g 2 S 1 g for i = n + 1. Excercise For a positive integer k, consider the Hirzebruch surface W (k) := f([a b] [x y z] 2 C P 1 C P 2 j a k y = b k xg 15

16 with the T -action dened by ([a b] [x y z])! ([g 1 a b] [g k 1 x y g 2z]): (1) Check that W (k) is a toric manifold. (2) Find four characteristic submanifolds W (k) i and the corresponding circle subgroups T i. (3) Find four T -xed points and the tangential T -representations at them. Construction of toric manifolds The orbit space of a toric manifold M is a convex polytope by denition. Note that M I =T is a codimension jij face of M=T. Now we reverse a gear. Namely we start with a convex polytope P and construct a toric manifold with P as the orbit space. To do this we need an extra data dened below. Let P denote an index set parametrizing facets of P, i.e., a facet of P will be denoted by P i for i 2 P. For a subset I of P, we denote by P I the intersection \ i2i P i. Note that jij n if P I 6=. Suppose that we are given a map F : P! H 2 (BT) such that if P I 6=, then ff(i) j i 2 Ig extends to a Z-basis of H 2 (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 follows. When P I 6=, let T I be the jij-dimensional subtorus generated by circle subgroups of T determined by F(i)'s (i 2 I). (Remember that a nonzero element in H 2 (BT) determines a circle subgroup of T.) We dene an equivalence relation on T P : (g 1 p 1 ) (g 2 p 2 ) if and only if p 1 = p 2 and g ;1 g T I where I is a subset of P such that P I contains the point p 1 = p 2 in its relative interior. One checks that the quotient space T P=, denoted by M(P F), is a manifold and the action of T by left translations descends to an action of T on M(P F), so that M(P F) is the desired toric manifold with P as the orbit space. Excercise Take P = [0 1] and construct S 2 by nding a characteristic map F. The projection onto the second factor induces a map q : M(P F)! P: Excercise Check that the characteristic submanifolds of M(P F) are given by q ;1 (P i ) (i 2 P ). Therefore we may assume that P = M(P F), and then q ;1 (P I ) = M(P F) I for any subset I P. In particular, T -xed pionts in M(P F) correspond to vertices in P via the map q. Excercise Observe that the maximal subtorus which xes M(P F) I pointwise is T I generated by circle subgroups corresponding to F(i)'s (i 2 I). 16

17 Example Take n = 2 and let P be a d-gon, that is a two-dimensional convex polytope with d sides. The facets of P are the sides. We number them as P 1 P 2 : : : P d in such a way that P i;1 and P i are adjacent for i = 1 : : : d, where P 0 = P d. Let v 1 : : : v d be a sequence of elements in H 2 (BT) = Z 2 such that each successive pair v i;1 and v i is a basis of H 2 (BT) for i = 1 : : : d, where v 0 = v d. We then dene F(i) = v i for each i. Excercise When d = 3 in the example above, prove that M(P F) is dieomorphic to C P 2. x6. Equivariant cohomology of toric manifolds. Throughout this section, M is a toric manifold. Cohomology of toric manifolds Denote by i 2 H 2 (M) the Poincare dual of a characteristic submanifold M i of M. Lemma 6.1. A toric manifold M is simply connected (hence, orientable), and H (M) is generated by i 's as a ring. Proof. Take a height function on P = M=T such that takes distinct values on vertices of P, and decompose M using into cells as follows. We orient each edge of P so that 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 F v be the smallest face of P which contains the inward pointing edges incident to v. Clearly, dim F v = m(v). We delete from F v all faces not incident to v. The resulting space ^Fv is dieomorphic to R m(v) +. Let e v be the inverse image of ^Fv by the projection map q : M! P. Since ^Fv is dieomorphic to R m(v) +, e v is dieomorphic to R 2m(v). (See Lemma 4.7.) The sets e v 's for all vertices v give a cell decomposition of M. Since each cell is even dimensional, we have that M is simply connected, H (M) has no torsion and H odd (M) = 0. More precisely, the closure of a cell of dimension less than 2n is an intersection of characteristic submanifolds M i 's because q ;1 (P i ) = M i and F v is an intersection of P i 's. This means that H (M) is generated by i 's as a ring. There are many relations among (products of) i 's, which will be determined at the end of this section. The lemma above together with Lemma 3.4 implies Corollary 6.2. H T (M) = H (BT) H (M) as H (BT)-modules. Simplicial complex ; M 17

18 As observed in Excercise 5.12, the T -action is free on Mn [ i2m M i, so one may expect that the characteristic properties of the action would be concentrated on (neighborhoods of) the M i 's. Using M i 's, we dene a combinatorial invariant ; M := fi M j M I 6= g: It describes how M i 's intersect. One checks that ; M is a simplicial complex of dimension n ; 1. Note that since M I 6= if and only if (M=T) I 6=, ; M is determined by the orbit space M=T. Example 6.3. If M is C P n with the T -action in Example 5.8, then ; M consists of all proper subsets of M = f1 : : : n + 1g so the geometric realization of ; M is the boundary of the standard n-simplex, which is homeomorphic to S n;1. If I 2 ; M, then it is a dimension jij;1 face of ; M while (M=T) I is a codimension jij ; 1 face This implies that ; M is the \dual complex" HT (M) and a face ring We shall see that the simplicial complex ; M is related to the ring structure of H T (M). Since M is orientable, so are M i's. We choose orientations on M and M i 's, and x them. Then the equivariant Gysin homomorphism is dened for the inclusion map f i : M i! M: We set f i! : H q T (M i)! H q+2 T (M): i := f i! (1) 2 H 2 T (M) Q where 1 2 HT 0 (M i) is the identity. For a subset I of M, we abbreviate i2i i as I. Geometrically, i is the Poincare dual of M i in equivariant cohomology, so I is the Poincare dual of M I. Therefore I must be zero whenever M I is empty, i.e., whenever I =2 ; M. It turns out that there is no other relation in HT (M). Namely we have Theorem 6.4 (see [DJ],[Ma]). Let R be a polynomial ring in variables x i 's (i 2 M ) and let I be the ideal in R generated by monomials x I 's for I =2 ; M. Then the map : R=I! H T (M) sending x i to i is a ring isomorphism. The ring R=I is determined by ; M and called the face ring (or Stanley-Reisner ring) of the simplicial complex ; M. (See [St].) We derive a combinatorial invariant from ; M. For 0 k n ; 1, let f k (M) be the number of k-simplicies in ; M. Dene a polynomial M(t) of degree n by M(t) = (t ; 1) n + n;1 X k=0 f k (M)(t ; 1) n;1;k and let h k (M) for 0 k n be the coecient of t n;k in M(t), i.e., M(t) = nx k=0 18 h k (M)t n;k :

19 Excercise 6.5. Find M(t) for the toric manifolds in Examples 5.8 and 5.9. Excercise 6.6. Prove that M(t 2 ) = P 2n q=0 b q(m)t q of M, where b q (M) denotes the q-th Betti number of M. Excercise 6.7. Prove that the Euler-Poincare characteristic P 2n q=0 (;1)q b q (M) of M agrees with the number of vertices in the simple convex polytope dieomorphic to M=T. Equivariant Pontrjagin classes of toric manifolds For x 2 M T, denote by I(x) the maximal subset of M such that x 2 M I(x). Note that ji(x)j = n. By denition i = f! (1) where f i : M i! M is the inclusion. Its restriciton to x, denoted by i j x, vanishes unless x 2 M i (in other words, unless i 2 I(x)), which follows from Property 3 of the equivariant Gysin homomorphism stated in section 3. Theorem 6.8. p T (M) = Q i2m (1 + 2 i ) in H T (M). Proof. By Corollary 3.6 it suces to prove that the restrictions of the both sides at the identity above to any point x 2 M T coincide. The restriction Q of the left hand side to x is p T ( x M) while that of the right hand side is (1 + i2i(x) 2 i j x). Thus it suces to prove that Y (6.9) p T ( x M) = (1 + i 2 j x): i2i(x) Since M i 's (i 2 I(x)) intersect transversally at x, we have x M = i2i(x) i j x as T -representations, where i j x denotes the ber over x of the normal bundle i of M i in M. Taking the equivariant Pontrjagin classes on both sides, we obtain p T ( x M) = Y i2i(x) p T ( i j x ): Here p T ( i j x ) is equal to the restriction of p T ( i ) to x, and since i is of dimension 2, we have p T ( i ) = 1 + p T 1 ( i ) = 1 + e T ( i ) 2 = 1 + f i ( 2 i ) where the second identity follows from [MS, Corollary 15.8] and the third identity follows from Property 4 of the equivariant Gysin homomorphism stated in section 3. Since f i (2 i )j x = 2 i j x, these identities prove the desired identity (6.9). Excercise Show that the set f i j x j i 2 I(x)g is a Z-basis of H 2 (BT). 19

20 Corollary Let i be the Poincare dual of the homology class in H 2n;2 (M) represented by M i. Then Y p(m) = (1 + 2 i ) in H (M): i2m Proof. Restrict the identity in Theorem 6.8 to H (M). Since i restricts to i, the corollary follows. Example Let M be C P n as in Example 5.8. Then M = f1 : : : n +1g and each i is equal to a generator of H 2 (C P n ) up to sign. It follows that p(c P n ) = (1 + 2 ) n+1 in H (C P n ), which is a well-known formula (see [MS, x15]). The algebra structure of H T (M) We have observed that the ring structure of HT (M) is determined by ; M. Since ; M is determined by the orbit space M=T, the ring structure of HT (M) is determined by M=T and has nothing to do with a characteristic map F. However, HT (M) has a ner structure than the ring structure, that is, an algebra structure over H (BT) through : H (BT)! HT (M). It turns out that the characteristic map comes in for the algebra structure. We have to investigate to know the algebra structure of HT (M) over H (BT). To see, it suces to see the image of elements in H 2 (BT) through because H (BT) is generated by elements in H 2 (BT) as a ring. Lemma There is a unique element v i 2 H 2 (BT) for each i 2 M such that X (u) = hu v i i i in HT 2 (M) for any u 2 H 2 (BT) i2m where h i denotes the natural pairing between cohomology and homology. Proof. Since HT 2 (M) is additively generated by i's by Theorem 6.4, (u) can be expressed uniquely as X (u) = v i (u) i i2m with integers v i (u) depending on u. We view v i (u) as a function on H 2 (BT). Clearly the function is linear, so it denes an element of Hom(H 2 (BT) Z) = H 2 (BT), that is our desired v i. Example Consider C P n with the T -action in Example 5.8. We give C P n and (C P n ) i (see Example 5.13) the orientations induced from the complex structures. Then v i = (0 : : : 1 : : : 0) (1 sits in the i-th place) for i = 1 : : : n and v n+1 = (;1 : : : ;1) when H 2 (BT) is naturally identied with Z n. Excercise Conrm Example (Hint: Look at the restrictions of the identity in Lemma 6.13 to T -xed points with the v i 's in Example 6.14.) 20

21 Lemma For any x 2 M T, the set fv i j i 2 I(x)g is the dual basis of f i j x j i 2 I(x)g. Proof. Take u = j j x for j 2 I(x) at the identity in Lemma 6.13 and restrict it to x. It reduces to X j j x = h j j x v i i i j x i2i(x) because i j x = 0 unless i 2 I(x). This together with Excercise 6.10 implies the lemma. Excercise Prove that the circle subgroup determined by v i is T i which xes M i pointwise. Dene F M : M! H 2 (BT) by F(i) = v i for i 2 M. Lemma 6.16 tells us that F M is a characteristic map. Excercise Observe that M is equivariantly dieomorphic to M(M=T F M ). If M = M(P F) with a charcterictic map F : P! H 2 (BT), then M = P and F(i) = F M (i) for each i 2 P = M up to sign. The sign depends on the choice of an orientation on M i. One can always choose an orientation on M i so that F(i) = F M (i) because F M (i) = v i changes into ;v i if we reverse the orientation on M i. Cohomology of toric manifolds revisited Since H odd (M) = 0, the spectral sequence of the bration M! M T! BT collapses so the restriction map: HT (M)! H (M) is surjective. Clearly, the ideal K generated by elements in H (BT) of positive degrees map to zero via the restriction map. In fact, one checks that the kernel of the restriction map is exactly the ideal K. Since positive degree elements in H (BT) are generated by elements in H 2 (BT), it follows from Theorem 6.4 and Lemma 6.13 that Theorem 6.19 ([DJ]). Let R be the polynomial ring in variables x i 's (i 2 M ) as before and let ~ I be the ideal in R generated by all (1) monomials x I (I =2 ; M ), (2) P i2m hu v iix i for u 2 H 2 (BT). Then the map : R=~ I! H (M) sending x i to i is a ring isomorphism. Excercise Check that H (C P n ) is a truncated polynomial ring Z[]=( n+1 ) using Theorem 6.19, where 2 H 2 (C P n ). 21

22 x7. Unitary toric manifolds and multi-fans. Unitary toric manifolds As observed in the last section, orientations on M and M i 's are necessary to dene the characteristic map F M. When M is a compact nonsingular toric variety, M and M i '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 M i 's. To get around this, we impose a complex structure on the tangent bundle M 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 S 2n admits a unitary structure for any n but admits an almost complex structure only when n = 1 or 3. Denition 7.1. We call a toric manifold M a unitary toric manifold if M R 2k for some k is endowed with a complex structure preserved by the T -action, where R 2k denotes the product bundle M R 2k! M. When k = 0, a unitary toric manifold is especially called an almost complex toric manifold. (The word \unitary (or almost complex) toric manifold" is used in [Ma] for a rather more general family of unitary manifolds with T -actions.) Henceforth M will denote a unitary toric manifold unless otherwise stated. Then M R 2k is oriented as a complex vector bundle and R 2k is oriented in the usual way. These orientations determine an orientation on M. Excercise 7.2. 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 M R 2k. By the excercise above, each M i is unitary, so it has an orientation induced from the unitary structure. Thus M and M i 's have canonical orientations. The tangential T -representation x M at x 2 M T is a complex representation since it is the nontrivial part of a complex T -representation (M R 2k ) x. In fact, we have X x M = i j x as complex T -representations. i2i(x) We dene the equivariant Chern class c T (M) or the Chern class c(m) of M to be c T (M R 2k ) or c(m R 2k ) respectively. The same argument as in Theorem 6.8 and Corollary 6.11 proves Theorem 7.3. c T (M) = Q i2m (1 + i) and c(m) = Q i2m (1 + i): Todd genus 22