Cohomology, symmetry, and perfection

Publicacions Matemátiques, Vol 36 (1992), 407-420 . A bstract 2 . 3 . COHOMOLOGY, SYMMETRY, AND PERFECTION EMILI BIFET * We explain the philosophy behind the computations in [BDP] and place them in a wider conceptual setting . We also outline, for toric varieties, the resulting equivariant approach to some key results in that theory . 1 . Symmetry In many situations that arise in Algebraic Geometry one is interested in computing the multiplicative structure of the cohomology ring H* (X), with rational coefficients say, of some algebraic variety X . Examples of such situations include toric varieties, complete quadrics, complete symmetric varieties [DP1-2], . . . Sometimes, as in the examples just mentioned, the variety X is endowed with symmetries that reflect the action of some algebraic group G on it . In these cases there is a recipe, inspired by the work of M . F . Atiyah and R . Bott [AB1], that often works : Find a strongly G-perfect decomposition of X (cf . Section 3 below for a precise statement .) In the examples above, this is simply the decomposition into orbits . In general there is a natural candidate : the Kempf-Hesselink stratification of X [H], [K], [N], a natural outgrowth of D . Mumford's Geometric Invariant Theory . With the help of this decomposition compute the equivariant cohomology ring HG(X) . (At this point it may also be natural to apply the machinery of the localization theorem [AB2], [Hs] ; in so doing one usually obtains other interesting descriptions of Hc(X) . ) Recover H* (X) from HG (X) . *This paper is dedicated to the memory of my friend Pere Menal . I chose the present topic for this occasion because it was the subject of our last conversation . I miss him very much . 40 8  E . BIFET This recipe is just one more instance of the old philosophy of using any symmetries that may be present in the problem in order to simplify it . We shall work, for simplicity, with equivariant cohomology defined in terms of homotopy quotients Le . the Borel construction (see Section 2 below .) But the knowledgeable reader could substitute througliout HG(X) by the smooth ~-adic cohomology of the algebraic stack determined by the G-variety X . He would thus gain the advantage of having a completely algebraic theory with, as a bonus, an interesting arithmetic twist [B2] . In the examples mentioned earlier, the recipe works and gives very explicit results . We shall consider below in some detail the case of toric varieties, but the reader is advised to look at [BDP] for a thorough treatment of three examples from the present point of view . In fact one of the aims of this paper is to better explain the philosophy behind those computations and to place them in a wider conceptual setting . Another aim of the paper is to outline in the last section an "equivariant" approach to some key results in the theory of toric varieties . This approach clarifies, I believe, the nature of three results . The text of the first three sections follows closely a talk delivered at the University of Copenhagen in July 1 .989 on the occasion of the Zeuthen Symposium . I would like to thank S . Kleiman and A . Thorup for organizing that conference and creating a very friendly athmosphere . 2 . Cohomology Suppose that a topological group G acts properly on a space X . The (equivariant) cohomology of the G-space X should be the cohomology of the quotient X/G . Unfortunately, unless G is acting freely on X, the usual topological quotient does not provide a useful theory . In general, it is necessary to find the right notion of quotient . The best notion is probably obtained by taking the quotient in the 2-category of toposes [SGA4] . However, for simplicity, we shall work here with a homotopy quotient XG . ( In a purely algebraic context the role of XG would be played by the algebraic stack determined by the G-variety X ; the equivariant cohomology would just be the smooth ~-adic cohomology of this stack . ) Before we can describe this notion of quotient, however, it is necessary to look closely at the case where X is a point Le . the theory of characteristic classes . Recall that a classifying space for principal G-bundles is by definition a space BG together with a universal principal G-bundle EG over it . By universal we mean that isomorphism classes of principal G-fibre bundles COHOMOLOGY, SYMMETRY, AND PERFECTION  409 over a nice space X correspond naturally to homotopy classes of maps from X to BG . The correspondence is given by pulling-back the universal bundle EG . The cohomology ring H* (BG) is by definition the ring of characteristic classes of G . Example 1 . G= C' . This is simply the theory of line bundles, and the classifying space is the infinite projective space PC . Its cohomology ring H*(BG) = Z[cl] is a polynomial ring in one variable of degree two . Example 2 . G=T = C lx . . . x C" (an algebraic torus .) In this case we have : BTBC x x . . . x BC x and its cohomology is a polynomial algebra in several variables (as many as factors .) In fact, if X(T) denotes the group of algebraic characters of T, then H*(BT) - Sym*X(T) . In particular H 2 (BT) - X (T) . Example 3 . G = GL  (C) . The classifying space is the infinite dimensional Grassmannian and H* (BG) = Z[cl, . . . , c,,] where the variables e2, 1 <_ i <_ n, have degree 2i and correspond to the Chern classes . In general, if T is a maximal torus in G, we have ( with rational coefficients ) H* (BG) =H*(BT ) W where W = NG(T)/T is the Weyl group of (G, T) and the right hand side denotes the subring of invariants . We are now ready to describe the homotopy quotient mentioned earlier . This is given by the Borel construction XG obtained after exchanging the fibre G of the universal bundle EG with X Le . XG=EGxGX=(EGxX)/G where G acts by g - (e, x) = (eg -1 , gx) . Definition . The equivariant cohomology HG(X) is by definition the cohomology of the Borel construction XG . Note that there is a fibration (2 .1)  X -+ XG --> BG . 410  E . BIFET The spectral sequence of this fibration is the key te the theed step in the recipe . This is based on work of P . Deligne [De], V . A . Ginzburg [G], F . Kirwan [K], . . . Here follow some other properties of equivariant cohomology with rational coefficients [Hs], [AB2] : a) If G acts freely en X, then HG(X) = H*(XIG) . b) If T is a maximal torus in G and W = NG(T)/T is the Weyl group, then HG(X) = HT(X)w where the right hand side is the subring of W-invariants . c) If X has a single orbit, then HG(X) - H* (B H) where H is the stabilizer of any point . d) If K is a maximal compact subgroup of G, then Hc(X) = Hix(X) . One of the reasons equivariant cohomology is easier to compute than ordinary cohomology is that it has many more "points" . Let me try to explain this . Most succesful calculations of cohomology achieve their objective by expressing the cohomology of the space under consideration (e .g . projective space P') in terms of that of spaces for which it is already known (e .g . cells .) Ultimately, however, they reduce the computation te that of the cohomology of a point . If one thinks of ordinary cohomology as being the case G= 1 of the equivariant one, then it is clear that the points coincide with the orbits . Thus in the equivariant theory every orbit gives rise to a "point", and there are as many points as there are conjugacy classes of subgroups in G . The equivariant cohomology of such a point H is precisely the ring of H-characteristic classes Le . the cohomology of the classifying space of H . It follows that in the equivariant theory there is much more freedom of movement . Another important feature of equivariant cohomology is that there is a theory of equivariant Chern classes . A G-linearization of a vector bundle F over X is an action u : G x F -> F which is linear en the fibres and turns the projection 7r : F ---> X into a G-equivariant map Le . 7 r(g - x) = g - 7r(x) for every g E G and every xE F . Note that the homotopy quotient FG provides us with a vector bundle over XG . The equivariant Chern classes of (F, u) are by definition the Chern classes of FG . This takes a most simple form for a line bundle over an orbit . In this case the equivariant Chern class c(L, u) is determined by the (3 .2) COHOMOLOGY, SYMMETRY,AND PERFECTION  411 isotropy action (character) of the stabilizer on the fibre of L over the point . Actually these notions find their most natural formulation when expressed in terms of algebraic stacks . For example a G-linearized Gxmodule is simply a module for the structure sheaf of the algebraic stack determined by the G-variety X . 3 . Perfection Let X be a smooth complex algebraic variety, and let the algebraic group G act on X . Suppose S C X is a closed G-invariant smooth subvariety and let U = X -S be the complementary open set . Under these conditions, there is a long exact sequence (the equivariant ThomGysin sequence, see [AB1] for example ) . . .  HG 2codimS(S) is HG(X) - HG(U)  . . . Moreover the composite of the maps then, for example, we have HG 2codimS(S) HG(X) 1 restriction Hc(S) is multiplication by the Euler class e(NSIx) (= top Chern class in this context) of the normal bundle N s 1 x . If this long exact sequence splits into short exact sequences 0 -> HG 2codimS(S) _, Hc(X) _ HG(U) -> 0 bG(X) = bG(U) + bG 2codimS(S) and one can deduce the equivariant Betti numbers of X from those of S and U . In [AB1] Atiyah and Bott made the following fundamental observation : If e(NS I x ) is not a zero-divisor in the ring H* (S), then the morphisms s are injective and the long exact sequences (3 .1) split into short exact sequences (3 .2) . This motivates : 412  E . BIFET Definition . We say that a decomposition X= Si U . . .USN is strongly G-perfect if : 1) Each S i is both smooth and G-invariant . 2) For each k, Xk =S1 U . . . USk is an open subset of X . 3) For each pair (Sk, Xk), k > 1, the Euler class e(NS,_IX,) is a non-zero divisor . In [AB1] a decomposition is defined to be G-perfect if the long exact sequence determined by each pair (Sk,Xk) splits into short exact sequences . In this case one has an identity of equivariant Poincaré series It is clear that strongly G-perfect implies G-perfect . An immediate consequence of the definition is Proposition . If {S2}1<á<N is a strongly G-perfect decomposition of X, then for every partial union Xk the morphism induced by the restrictions to the strata (3 .3)  HG(Xk) - H Hc(SZ) 1<¡<k is injective . PC(x) _  t2 .codimSi . PG(S2) 1<i<N Proof .. For k = 1, it is obvious . Suppose it holds for k - 1 ; it sufflces to show that the morphsm are injective . Consider the diagram Hi-2codimSk (sk) G HG(Xk) - HG(Xk-1) x H * (Sk) HG(Xk-1 U sk)  HG(Xk-1) HG(Sk) - Proof . Since HT(P) . But, if COHOMOLOGY, SYMMETRY, AND PERFECTION  41 3 Now, if 71(a) = 0, then there is a b E HG 2codiMSk (Sk) such that a = ~(b) . But, from 0 = (a) = («b)) = b U e(NS, 1x k ) it follows that b = 0 and therefore a = « b) = 0 . a Thus, in principle, if ose knows the cohomology rings of the strata and one controls the injection above, it is possible to describe the cohomology ring of X . This is the reason we singled out this notion for special consideration . Atiyah and Bott also give an infinitesimal criterios for e(NSIx) to be a non-zero divisor (see [AB1 Proposition 13 .4 .]) It is proved in [K] using this criterios that the Kempf-Hesselink stratification [H] of a G-variety is strongly G-perfect in the above sense . Let us enunciate this last criterios in the case of an orbit : Proposition . Let C be a G-orbit in the smooth algebraic variety X . Choose a point P E C and identify C with G/Gp, where Gp is the stabilizer of P . If the isotropie action of a maximal torus T in Gp on the normal space NOIx(P) has no non-zero fixed points, then e(NoIx) is a non-zero divisor . Hc(0) = Hcp (P) - HT(P) is an embedding, it suffices to see that e = e(N(DIx) is non-zero in NoIx(P) =  ®  Cxi X¡EX(T) is the weight decomposition, then e=l1xi :7É o . Since all xz are non-zero . The considerations above motivate : Definition . We say that X is a perfect embedding (regular embedding in [BDP], but this was a bad choice to which I plead guilty) provided a) Each orbit closure C is smooth and it is the transversal intersection of the codimension one orbit closures that contain it . b) For every P E 0, the stabilizer Gp has a dense orbit in the normal space NOIX(P) . To any perfect embedding X we associate a simplicial complex Cx = (V, S) as follows : 41 4  E . BIFET 1 . V = {v 1 O v is an orbit of codimension one} . 2 . I' C V is a simplex if, and only if, vEr (Note that ? is a simplex .) O v7 ~ ? . It is easy to show that the simplexes are in one-to-one correspondence with the orbits . It is clear that the decomposition of X into orbits is in this case strongly G-perfect . The algebraic varieties mentioned above, namely toric varieties and complete symmetric varieties (in particular complete quadrics), provide examples of perfect embeddings . In [BDP] an explicit description, based on these ideas, is given for the equivariant cohomology ring of any perfect embedding . It should be possible to extend these results to the case of a well behaved G-variety and the Kempf-Hesselink stratification . 4 . An example : toric varieties We shall illustrate the generalities of the preceding sections with the concrete case of toric varieties . Let T be an algebraic torus . A toric variety is a normal algebraic T-variety X containing T as a dense orbit (this includes the requirement that the stabilizer at the points of this orbit be trivial .) Referentes [D], [F], [O] provide very good expositions of the basic theory of these varieties . A simple example is the affine plane with the action of T= C" x C X given by (t1,t2)(xl>x2) = (t1X1,t2x2) This action has four orbits, namely the origin, the two punctured axes and the torus T itself . This can be generalized to the action t - (x1, x2) = (tx1 x l, t x2 x2) determined by any integral basis {X1, X2} of the character group X (T) - Z 2 . This action induces an action on the ring of regular functions on the affine plane, the ring of polynomials in two variables, given by (t f ) (x) = f (t -1 x) . The weight functions (Le . the functions f such that t f = tX f for some character X called the weight of f) are precisely the monomials a-Xi 1 X2 2 and their weight is n1( - X1)+ n2 ( - X2) . These weights span a COHOMOLOGY, SYMMETRY, AND PERFECTION  41 5 cope av in X(T) ®R . Conversely we can recover the variety, including the T-action, by taking the Spectrum of the monoid algebra C[a v nX(T)] or, what is essentially the same, the algebra homomorphisms from C[av n X(T)] to C . In general, affine toric varieties can be constructed as follows . We denote Y(T) = Hom(C", T) the group of one-parameter subgroups of the algebraic torus T . (Recall that there is a pairing X(T) x Y(T) -~ Z given by taking (X, w) to be the unique integer such that X(M(t)) = t(X,w) for all t E C X .) First we consider a cone a = {Alfil + . . . +  1 A2 E R, Ai > 0 for every i} in Y(T)R= Y(T) ®Z R where {mi, . . . ,m  ,,} are finitely many oneparameter subgroups (actually we also ask that the cone o . have a vertex Le . a n (-a) = 0 .) Next we introduce its dual in X(T)R= X(T) OZR given by av = {X E X(T)R 1 (X, tt) > 0 for every mEa} . Then we express the monoid av n X(T) in terms of a finite number of generators (4 .1) o,vnX(T)=N-XI+ . . .+N-XN (that this can always be done is a consequence of Gordan's lemma [D], [F], [O] .) Finally we construct the affine model X Q by taking, as in the case of the affine plane, the Spectrum of the monoid algebra or equivalently the (scheme theoretic) closure of the image of the map T -+ CN sending t to (tXl , . . . , tXN ) . A toric variety is obtained by glueing together the affine models above along T-invariant open subsets . Think for example of the projective plane obtained by glueing there copies of the affine plane along the open orbit and identifying the punctured axes in pairs . Of course, here we are glueing not only the spaces but also the T-actions so the actions on our there affine planes have to be compatible . Fortunately the cones introduced earlier allow this compatibility to be expressed in simple terms . Definition . Let T be an algebraic torus . A collection E of cones as above in Y(T)R is said to be a fan whenever it satisfies the following properties a) Every face of a cone a in E also belongs to E . b) The intersection of any two cones in E is a face of both of them .