EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY
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10% de réduction sur vos envois d'emailing --> CLIQUEZ ICI Retour à l'accueil, cliquez ici EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY IOANID ROSU Abstract. Equivariant elliptic cohomology with complex coe?cients was de?ned axiomatically by Ginzburg, Kapranov and Vasserot [9] and constructed by Grojnowski [10]. We give an invariant de?nition of complex S 1 -equivariant elliptic cohomology, and use it to give an entirely cohomological proof of the rigidity theorem of Witten for the elliptic genus. We also state and prove a rigidity theorem for families of elliptic genera. Contents 1. Introduction 1 2. Statement of results 2 3. S 1 -equivariant elliptic cohomology 4 4. S 1 -equivariant elliptic pushforwards 9 5. Rigidity of the elliptic genus 14 Appendix A. Equivariant characteristic classes 22 References 25 1. Introduction The classical level 2 elliptic genus is de?ned (see Landweber [14], p.56) as the Hirzebruch genus with exponential series the Jacobi sine 1 . It is intimately related with the mysterious ?eld of elliptic cohomology (see Segal [19]), and with string theory (see Witten [22] and [23]). A striking property of the elliptic genus is its rigidity with respect to group actions. This was conjectured by Ochanine in [18], and by Witten in [22], where he used string theory arguments to support it. Rigorous mathematical proofs for the rigidity of the elliptic genus were soon given by Taubes [21], Bott & Taubes [4], and Liu [15]. While Bott and Taubes’s proof involved the localization formula in equivariant K-theory, Liu’s proof focused on the modularity properties of the elliptic genus. The question remained however whether one could ?nd a direct connection between the rigidity theorem and elliptic cohomology. Earlier on, Atiyah and Hirzebruch [2] had used pushforwards in equivariant K-theory to prove the rigidity of the Aˆ -genus for spin manifolds. Following this idea, H. Miller [16] interpreted the equivariant elliptic genus as a pushforward in the completed Borel equivariant cohomology, and posed the problem of developing and using a noncompleted S 1 -equivariant elliptic cohomology, to prove the rigidity theorem. In 1994 Grojnowski [10] proposed a noncompleted equivariant elliptic cohomology theory with complex coe?cients. For G a compact connected Lie group he de?ned E * G(-) as a coherent holomorphic sheaf over a certain variety XG constructed from a given elliptic curve. 1 For a de?nition of the Jacobi sine s(x) see the beginning of Section 4. 12 IOANID ROSU Grojnowski also constructed pushforwards in this theory. At about the same time and independently, Ginzburg, Kapranov and Vasserot [9] gave an axiomatic description of equivariant elliptic cohomology. Given Grojnowski’s construction, it seemed natural to try to use S 1 -equivariant elliptic cohomology to prove the rigidity theorem. In doing so, we noticed that our proof relies on a generalization of Bott and Taubes’ “transfer formula” (see [4]). This generalization turns out to be essentially equivalent to the existence of a Thom class (or orientation) in S 1 -equivariant elliptic cohomology. We can generalize the results of this paper in several directions. One is to extend the rigidity theorem to families of elliptic genera, which we do in Theorem 5.6. Another would be to generalize from G = S 1 to an arbitrary connected compact Lie group, or to replace complex coe?cients with rational coe?cients for all cohomology theories involved. Such generalizations will be treated elsewhere. 2. Statement of results All the cohomology theories involved in this paper have complex coe?cients. If X is a ?nite S 1 -CW complex, H* S1 (X) denotes its Borel S 1 -equivariant cohomology with complex coe?cients (see Atiyah and Bott [1]). If X is a point *, H* S1 (*) =~ C[u]. Let E be an elliptic curve over C. Let X be a ?nite S 1 -CW complex, e.g. a compact S 1 - manifold 2 . Then, following Grojnowski [10], we de?ne E * S1 (X), the S 1 -equivariant elliptic cohomology of X. This is a coherent analytic sheaf of Z2-graded algebras over E. We alter his de?nition slightly, in order to show that the de?nition of E * S1 (X) depends only on X and the elliptic curve E. Let a be a point of E. We associate a subgroup H(a) of S 1 as follows: if a is a torsion point of E of exact order n, H(a) = Zn; otherwise, H(a) = S 1 . We de?ne Xa = XH(a) , the subspace of X ?xed by H(a). Then we will de?ne a sheaf E * S1 (X) over E whose stalk at a is E * S1 (X)a = H* S1 (Xa ) ?C[u] OC,0 . Here OC,0 represents the local ring of germs of holomorphic functions at zero on C = Spec C[u]. In particular, the stalk of E * S1 (X) at zero is H* S1 (X) ?C[u] OC,0. THEOREM A. E * S1 (X) only depends on X and the elliptic curve E. It extends to an S 1 -equivariant cohomology theory with values in the category of coherent analytic sheaves of Z2-graded algebras over E. If f : X ? Y is a complex oriented map between compact S 1 -manifolds, Grojnowski also de?nes equivariant elliptic pushforwards. They are maps of sheaves of OE-modules f E ! : E * S1 (X) [f ] ? E * S1 (Y ) satisfying properties similar to those of the usual pushforward (see Dyer [7]). E * S1 (X) [f ] has the same stalks as E * S1 (X), but the gluing maps are di?erent. If Y is a point, then f E ! (1) on the stalks at zero is the S 1 -equivariant elliptic genus of X (which is a power series in u). By analyzing in detail the construction of f E ! , we obtain the following interesting result, which answers a question posed by H. Miller and answered independently by Dessai and Jung [6]. PROPOSITION B. The S 1 -equivariant elliptic genus of a compact S 1 -manifold is the Taylor expansion at zero of a function on C which is holomorphic at zero and meromorphic everywhere. 2 A compact S 1 -manifold always has an S 1 -CW complex structure: see Alday and Puppe [3].EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 3 Grojnowski’s construction raises a few natural questions. First, can we say more about E * S1 (X) [f ] ? The answer is given in Proposition 5.7, where we show that, up to an invertible sheaf, E * S1 (X) [f ] is the S 1 -equivariant elliptic cohomology of the Thom space of the stable normal bundle to f. (In fact, if we enlarge our category of equivariant CW-complexes to include equivariant spectra, we can show that E * S1 (X) [f ] is the reduced E * S1 of a Thom spectrum X-T f . See the discussion after Proposition 5.7 for details.) This suggests looking for a Thom section (orientation) in E * S1 (X) [f ] . More generally, given a real oriented vector bundle V ? X, we can twist E * S1 (X) in a similar way to obtain a sheaf, which we denote by E * S1 (X) [V ] . For the rest of this section we regard all the sheaves not on E, but on a double cover E˜ of E. The reason for this is given in the beginning of Subsection 5.2. So when does a Thom section exist in E * S1 (X) [V ] ? The answer is the following key result. THEOREM C. If V ? X is a spin S 1 -vector bundle over a ?nite S 1 -CW complex, then the element 1 in the stalk of E * S1 (X) [V ] at zero extends to a global section, called the Thom section. The proof of Theorem C is essentially a generalization of Bott and Taubes’ “transfer formula” (see [4]). Indeed, when we try to extend 1 to a global section, we see that the only points where we encounter di?culties are certain torsion points of E which we call special (as de?ned in the beginnning of Section 3). But extending our section at a special point a amounts to lifting a class from H* S1 (XS 1 ) ?C[u] OC,0 to H* S1 (Xa ) ?C[u] OC,0 via the restriction map i * : H* S1 (Xa ) ?C[u] OC,0 ? H* S1 (XS 1 ) ?C[u] OC,0. This is not a problem, except when we have two di?erent connected components of XS 1 inside one connected component of Xa . Then the two natural lifts di?er up to a sign, which can be shown to disappear if V is spin. This observation is due to Bott and Taubes, and is the centerpiece of their “transfer formula.” Given Theorem C, the rigidity theorem of Witten follows easily: Let X be a compact spin S 1 -manifold. Then the S 1 -equivariant pushforward of f : X ? * is a map of sheaves f E ! : E * S1 (X) [f ] ? E * S1 (*). From the discussion after Theorem A, we know that on the stalks at zero f E ! (1) is the S 1 -equivariant elliptic genus of X, which is a priori a power series in u. Theorem C with V = T X says that 1 extends to a global section in E * S1 (X) [f ] = E * S1 (X) [T X] . Therefore f E ! (1) is the germ of a global section in E * S1 (*) = OE. But any such section is a constant, so the S 1 -equivariant elliptic genus of X is a constant. This proves the rigidity of the elliptic genus (Corollary 5.5). Now the greater level of generality of Theorem C allows us to extend the rigidity theorem to families of elliptic genera. The question of stating and proving such a theorem was posed by H. Miller in [17]. THEOREM D. (Rigidity for families) Let p : E ? B be a spin oriented S 1 -equivariant ?bration. Then the elliptic genus of the family p E ! (1) is constant as a rational function, i.e. when the generator u of H* S1 (*) = C[u] is inverted.4 IOANID ROSU 3. S 1 -equivariant elliptic cohomology In this section we give the construction of S 1 -equivariant elliptic cohomology with complex coe?cients. But in order to set up this functor, we need a few de?nitions. 3.1. Definitions. Let E be an elliptic curve over C with structure sheaf OE. Let ? be a uniformizer of E, i.e. a generator of the maximal ideal of the local ring at zero OE,0. We say that ? is an additive uniformizer if for all x, y ? V? such that x + y ? V?, we have ?(x + y) = ?(x) + ?(y). An additive uniformizer always exists, because we can take for example ? to be the local inverse of the group map C ? E, where the universal cover of E is identi?ed with C. Notice that any two additive uniformizers di?er by a nonzero constant, because the only additive continuous functions on C are multiplications by a constant. Let V? be a neighborhood of zero in E such that ? : V? ? C is a homeomorphism on its image. Denote by ta translation by a on E. We say that a neighborhood V of a ? E is small if t-a(V ) ? V?. Let a ? E. We say that a is a torsion point of E if there exists n > 0 such that na = 0. The smallest n with this property is called the exact order of a. Let X be a ?nite S 1 -CW complex. If H ? S 1 is a subgroup, denote by XH the submanifold of X ?xed by each element of H. Let Zn ? S 1 be the cyclic subgroup of order n. De?ne a subgroup H(a) of S 1 by: H(a) = Zn if a is a torsion point of exact order n; H(a) = S 1 otherwise. Then denote by Xa = XH(a) . Now suppose we are given an S 1 -equivariant map of S 1 -CW complexes f : X ? Y . A point a ? E is called special with respect to f if either Xa =6 XS 1 or Y a =6 Y S 1 . When it is clear what f is, we simply call a special. A point a ? E is called special with respect to X if it is special with respect to the identity function id : X ? X. An indexed open cover U = (Ua)a?E of E is said to be adapted (with respect to f) if it satis?es the following conditions: 1. Ua is a small open neighborhood of a; 2. If a is not special, then Ua contains no special point; 3. If a =6 a 0 are special points, Ua n Ua0 = Ø. Notice that, if X and Y are ?nite S 1 -CW complexes, then there exists an open cover of E which is adapted to f. Indeed, the set of special points is a ?nite subset of E. If X is a ?nite S 1 -CW complex, we de?ne the holomorphic S 1 -equivariant cohomology of X to be HO * S1 (X) = H* S1 (X) ?C[u] OC,0 . OC,0 is the ring of germs of holomorphic functions at zero in the variable u, or alternatively it is the subring of C[[u]] of convergent power series with positive radius of convergence. Notice that HO* S1 is not Z-graded anymore, because we tensored with the inhomogenous object OC,0. However, it is Z2-graded, by the even and odd part, because C[u] and OC,0 are concentrated in even degrees. 3.2. Construction of E * S1 We are going to de?ne now a sheaf F = F?,U over E whose stalk at a ? E is isomorphic to HO* S1 (Xa ). Recall that, in order to give a sheaf F over a topological space, it is enough to give an open cover (Ua)a of that space, and a sheaf Fa on each Ua together with isomorphismsEQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 5 of sheaves faß : Fa|UanUß -? Fß|UanUß , such that faa is the identity function, and the cocycle condition fß?faß = fa? is satis?ed on Ua n Uß n U? . Fix ? an additive uniformizer of E. Consider an adapted open cover U = (Ua)a?E. De?nition 3.1. De?ne a sheaf Fa on Ua by declaring for any open U ? Ua Fa(U) := H* S1 (Xa ) ?C[u] OE(U - a) . The map C[u] ? OE(U - a) is given by sending u to ? (the germ ? extends to U - a because Ua is small). U - a represents the translation of U by -a, and OE(U - a) is the ring of holomorphic functions on U - a. The restriction maps of the sheaf are de?ned so that they come from those of the sheaf OE. First we notice that we can make Fa into a sheaf of OE |Ua -modules: if U ? Ua, we want an action of f ? OE(U) on Fa(U). The translation map ta : U - a ? U, which takes u to u + a gives a translation t * a : OE(U) ? OE(U - a), which takes f(u) to f(u + a). Then we take the result of the action of f ? OE(U) on µ ? g ? Fa(U) = H* S1 (Xa ) ?C[u] OE(U - a) to be µ ? (t * af · g). Moreover, Fa is coherent because H* S1 (Xa ) is a ?nitely generated C[u]-module. Now for the second part of the de?nition of F, we have to glue the di?erent sheaves Fa we have just constructed. If Ua n Uß =6 Ø we need to de?ne an isomorphism of sheaves faß : Fa|UanUß -? Fß|UanUß which satis?es the cocycle condition. Recall that we started with an adapted open cover (Ua)a?E. Because of the condition 3 in the de?nition of an adapted cover, a and ß cannot be both special, so we only have to de?ne faß when, say, ß is not special. In that case Xß = XS 1 . Consider an arbitrary open set U ? Ua n Uß. De?nition 3.2. De?ne faß as the composite of the following maps: (*) Fa(U) = H* S1 (Xa ) ?C[u] OE(U - a) ? H* S1 (Xß ) ?C[u] OE(U - a) ? (H* (Xß ) ?C C[u]) ?C[u] OE(U - a) ? H* (Xß ) ?C OE(U - a) ? H* (Xß ) ?C OE(U - ß) ? (H* (Xß ) ?C C[u]) ?C[u] OE(U - ß) ? H* S1 (Xß ) ?C[u] OE(U - ß) = Fß(U) . The map on the second row is the natural map i * ? 1, where i : Xß ? Xa is the inclusion. Lemma 3.3. faß is an isomorphism. Proof. The second and and the sixth maps are isomorphisms because Xß = XS 1 , and therefore H* S1 (Xß ) ~ -? H* (Xß ) ?C C[u]. The properties of the tensor product imply that the third and the ?fth maps are isomorphisms. The fourth map comes from translation by ß - a, so it is also an isomorphism. Finally, the second map i * ? 1 is an isomorphism because a) If a is not special, then Xa = XS 1 = Xß , so i * ? 1 is the identity. b) If a is special, then Xa =6 Xß . However, we have (Xa ) S 1 = XS 1 = Xß . Then we can use the Atiyah–Bott localization theorem in equivariant cohomology from [1]. This says that i * : H* S1 (Xa ) ? H* S1 (Xß ) is an isomorphism after inverting u. So it is enough to show that ? is invertible in OE(U - a), because this would imply that i * becomes an isomorphism after tensoring with OE(U - a) over C[u]. Now, because a is special, the condition 2 in the de?nition of an adapted cover says that a /? Uß. But6 IOANID ROSU U ? Ua n Uß, so a /? U, hence 0 ?/ U - a. This is equivalent to ? being invertible in OE(U - a). Remark 3.4. To simplify notation, we can describe faß as the composite of the following two maps: H* S1 (Xa ) ?C[u] OE(U - a) i * -? H* S1 (Xß ) ?C[u] OE(U - a) t * ß-a -? H* S1 (Xß ) ?C[u] OE(U - ß) . By the ?rst map we really mean i * ? 1. The second map is not 1 ? t * ß-a , because t * ß-a is not a map of C[u]-modules. However, we use t * ß-a as a shorthand for the corresponding composite map speci?ed in (*). Note that faß is linear over OE(U), so we get a map of sheaves of Z2-graded OE(U)-algebras. One checks easily now that faß satis?es the cocycle condition: Suppose we have three open sets Ua, Uß and U? such that Ua n Uß n U? =6 Ø. Because our cover was chosen to be adapted, at least two out of the three spaces Xa , Xß and X? are equal to XS 1 . Thus the cocycle condition reduces essentially to t * ?-ß t * ß-a = t * ?-a, which is clearly true. De?nition 3.5. Let U = (Ua)a?E be an adapted cover of E, and ? an additive uniformizer. We de?ne a sheaf F = F?,U on E by gluing the sheaves Fa from De?nition 3.1 via the gluing maps faß de?ned in 3.2. One can check now easily that F is a coherent analytic sheaf of algebras. Notice that we can remove the dependence of F on the adapted cover U as follows: Let U and V be two covers adapted to (X, A). Then any common re?nement W is going to be adapted as well, and the corresponding maps of sheaves F?,U ? F?,W ? F?,V are isomorphisms on stalks, hence isomorphisms of sheaves. Therefore we can omit the subscript U, and write F = F?. Next we want to show that F? is independent of the choice of the additive uniformizer ?. Proposition 3.6. If ? and ? 0 are two additive uniformizers, then there exists an isomorphism of sheaves of OE-algebras f?? 0 : F? ? F? 0 . If ? 00 is a third additive uniformizer, then f? 0 ? 00f?? 0 = ±f?? 00 . Proof. We modify slightly the notations used in De?nition 3.1 to indicate the dependence on ?: F ? a(U) = H* S1 (Xa ) ?? C[u] OE(U - a). Recall that u is sent to ? via the algebra map C[u] ? OE(U - a). If ? 0 is another additive uniformizer, we saw at the beggining of this Section that there exists a nonzero constant a in OE,0 such that ? = a? 0 . Choose a square root of a and denote it by a 1/2 . De?ne a map f?? 0 ,a : F ? a(U) ? F ? 0 a (U) by x ?? g 7? a |x|/2 x ?? 0 g. We have assumed that x is homogeneous in H* S1 (Xa ), and that |x| is the homogeneous degree of x. One can easily check that f?? 0 ,a is a map of sheaves of OE-algebras. We also have f ? 0 aß ? f?? 0 ,a = f?? 0 ,ß ? f ? aß , which means that the maps f?? 0 ,a glue to de?ne a map of sheaves f?? 0 : F? ? F? 0 . The equality f? 0 ? 00f?? 0 = ±f?? 00 comes from (? 0 /? 00 ) 1/2 (?/? 0 ) 1/2 = ±(?/? 00 ) 1/2 . De?nition 3.7. The S 1 -equivariant elliptic cohomology of the ?nite S 1 -CW complex X is the sheaf F = F?,U constructed above, which according to the previous results does not depend on the adapted open cover U or on the additive uniformizer ?. Denote this sheaf by E * S1 (X). If X is a point, one can see that E * S1 (X) is the structure sheaf OE.EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 7 Theorem 3.8. E * S1 (-) de?nes an S 1 -equivariant cohomology theory with values in the category of coherent analytic sheaves of Z2-graded OE-algebras. Proof. For E * S1 (-) to be a cohomology theory, we need naturality. Let f : X ? Y be an S 1 -equivariant map of ?nite S 1 -CW complexes. We want to de?ne a map of sheaves f * : E * S1 (Y ) ? E * S1 (X) with the properties that 1 * X = 1E* S1 (X) and (fg) * = g * f * . Choose U an open cover adapted to f, and ? an additive uniformizer of E. Since f is S 1 -equivariant, for each a we get by restriction a map fa : Xa ? Y a . This induces a map H* S1 (Y a ) ?C[u] OE(U - a) f * a?1 -? H* S1 (Xa ) ?C[u] OE(U - a). To get our global map f * , we only have to check that f * a glue well, i.e. that they commute with the gluing maps faß. This follows easily from the naturality of ordinary equivariant cohomology, and from the naturality in X of the isomorphism H* S1 (XS 1 ) =~ H* (XS 1 ) ?C C[u]. Also, we need to de?ne E * S1 for pairs. Let (X, A) be a pair of ?nite S 1 -CW complexes, i.e. A is a closed subspace of X, and the inclusion map A ? X is S 1 -equivariant. We then de?ne E * S1 (X, A) as the kernel of the map j * : E * S1 (X/A) ? E * S1 (*), where j : * = A/A ? X/A is the inclusion map. If f : (X, A) ? (Y, B) is a map of pairs of ?nite S 1 -CW complexes, then f * : E * S1 (Y, B) ? E * S1 (X, A) is de?ned as the unique map induced on the corresponding kernels from f * : E * S1 (Y ) ? E * S1 (X). Now we have to de?ne the coboundary map d : E * S1 (A) ? E *+1 S1 (X, A). This is obtained by gluing the maps H* S1 (Aa ) ?C[u] OE(U - a) da?1 -? H *+1 S1 (Xa , Aa ) ?C[u] OE(U - a), where da : H* S1 (Aa ) ? H *+1 S1 (Xa , Aa ) is the usual coboundary map. The maps da ? 1 glue well, because da is natural. To check the usual axioms of a cohomology theory: naturality, exact sequence of a pair, and excision for E * S1 (-), recall that this sheaf was obtained by gluing the sheaves Fa along the maps faß. Since Fa were de?ned using H* S1 (Xa ), the properties of ordinary S 1 -equivariant cohomology pass on to E * S1 (-), as long as tensoring with OE(U - a) over C[u] preserves exactness. But this is a classical fact: see for example the appendix of Serre [20]. This proves THEOREM A stated in Section 2. Remark 3.9. Notice that we can arrange our functor E * S1 (-) to take values in the category of coherent algebraic sheaves over E rather than in the category of coherent analytic sheaves. This follows from a theorem of Serre [20] which says that the the categories of coherent holomorphic sheaves and coherent algebraic sheaves over a projective variety are equivalent. 3.3. Alternative description of E * S1 For calculations with E * S1 (-) we want a description which involves a ?nite open cover of E. Start with an adapted open cover (Ua)a?E. Recall that the set of special points with respect to X is ?nite. Denote this set by {a1, . . . , an}. To simplify notation, denote for i = 1, . . . , n Ui := Uai , and U0 := E \ {a1, . . . , an} . On each Ui , with 0 = i = n, we de?ne a sheaf G as follows: a) If 1 = i = n, then ?U ? Ui , Gi(U) := H* S1 (Xai ) ?C[u] OE(U - ai). The map C[u] ? OE(U - ai) was described in De?nition 3.1. b) If i = 0, then ?U ? U0, Gi(U) := H* (XS 1 ) ?C OE(U).8 IOANID ROSU Now glue each Gi to G0 via the map of sheaves fi0 de?ned as the composite of the following isomomorphisms (U ? UinU0): H* S1 (Xai )?C[u]OE(U -ai) i *?1 -? H* S1 (XS 1 )?C[u]OE(U -ai) =~ -? H* (XS 1 ) ?C OE(U - ai) t * -ai -? H* (XS 1 ) ?C OE(U). Since there cannot be three distinct Ui with nonempty intersection, there is no cocycle condition to verify. Proposition 3.10. The sheaf G we have just described is isomorphic to F, thus allowing an alternative de?nition of E * S1 (X). Proof. One notices that U0 = ?{Uß | ß nonspecial}, because of the third condition in the de?nition of an adapted cover. If U ? ?ßUß, a global section in F(U) is a collection of sections sß ? F(U n Uß - ß) which glue, i.e. t * ß-ß0sß = sß0 . So t * -ß sß = t * -ß0sß0 in G(U n Uß n Uß0), which means that we get an element in G(U), since the Uß’s cover U. So F|U0 =~ G|U0 . But clearly F|Ui =~ G|Ui for 1 = i = n, and the gluing maps are compatible. Therefore F =~ G. As it is the case with any coherent sheaf of OE-modules over an elliptic curve, E * S1 (X) splits (noncanonically) into a direct sum of a locally free sheaf, i.e. the sheaf of sections of some holomorphic vector bundle, and a sum of skyscraper sheaves. Given a particular X, we can be more speci?c: We know that H* S1 (X) splits noncanonically into a free and a torsion C[u]-module. Given such a splitting, we can speak of the free part of H* S1 (X). Denote it by H* S1 (X)f ree. The map i *H* S1 (X)f ree ? H* S1 (XS 1 ) is an injection of ?nitely generated free C[u]-modules of the same rank, by the localization theorem. C[u] is a p.i.d., so by choosing appropriate bases in H* S1 (X)f ree and H* S1 (XS 1 ), the map i * can be written as a diagonal matrix D(u n1 , . . . , u nk ), ni = 0. Since i * 1 = 1, we can choose n1 = 0. So at the special points ai , we have the map i * : H* S1 (Xai )f ree ? H* S1 (XS 1 ), which in appropriate bases can be written as a diagonal matrix D(1, u n2 , . . . , u nk ). This gives over Ui n U0 the transition functions u 7? D(1, u n2 , . . . , u nk ) ? GL(n, C). However, we have to be careful since the basis of H* S1 (XS 1 ) changes with each ai , which means that the transition functions are diagonal only up to a (change of base) matrix. But this matrix is invertible over C[u], so we get that the free part of E * S1 (X) is a sheaf of sections of a holomorphic vector bundle. An interesting question is what holomorphic vector bundles one gets if X varies. Recall that holomorphic vector bundles over elliptic curves were classi?ed by Atiyah in 1957. Example 3.11. Calculate E * S1 (X) for X = S 2 (n) = the 2-sphere with the S 1 -action which rotates S 2 n times around the north-south axis as we go once around S 1 . If a is an n-torsion point, then Xa = X. Otherwise, Xa = XS 1 , which consists of two points: {P+, P-}, the North and the South poles. Now H* S1 (S 2 (n)) = H* (BS 1 ? BS 1 ) = C[u] ×C C[u], on which C[u] acts diagonally. i * : H* S1 (X) ? H* S1 (XS 1 ) is the inclusion C[u] ×C C[u] ,? C[u] × C[u]. Choose the bases a) {(1, 1), (u, 0)} of C[u] ×C C[u]; b) {(1, 1), (1, 0)} of C[u] × C[u]. Then H* S1 (X) ~ -? C[u] ? C[u] by (P (u), Q(u)) 7? (P, Q-P u ), and H* S1 (XS 1 ) ~ -? C[u] ? C[u] by (P (u), Q(u)) 7? (P, Q - P ). Hence i * is given by the diagonal matrix D(1, u). So E * S1 (X) looks locally like OCP 1 ? OCP 1 (-1 · 0). This happens at all the n-torsion points of E, so E * S1 (X) =~ OE ? OE(?), where ? is the divisor which consists of all n-torsion points of E, with multiplicity 1.EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 9 One can also check that the sum of all n-torsion points is zero, so by Abel’s theorem the divisor ? is linearly equivalent to -n 2 · 0. Thus E * S1 (S 2 (n)) =~ OE ? OE(-n 2 · 0). We stress that the decomposition is only true as sheaves of OE-modules, not as sheaves of OE-algebras. Remark 3.12. Notice that S 2 (n) is the Thom space of the S 1 -vector space C(n), where z acts on C by complex multiplication with z n . This means that the Thom isomorphism doesn’t hold in S 1 -equivariant elliptic cohomology, because E * S1 (*) = OE, while the reduced S 1 -equivariant elliptic cohomology of the Thom space is E˜ * S1 (S 2 (n)) = OE(-n 2 · 0). 4. S 1 -equivariant elliptic pushforwards While the construction of E * S1 (X) depends only on the elliptic curve E, the construction of the elliptic pushforward f E ! involves extra choices, namely that of a torsion point of exact order two on E, and a trivialization of the cotangent space of E at zero. 4.1. The Jacobi sine Let (E, P, µ) be a triple formed with a nonsingular elliptic curve E over C, a torsion point P on E of exact order two, and a 1-form µ which generates the cotangent space T * 0 E. For example, we can take E = C/?, with ? = Z?1 + Z?2 a lattice in C, P = ?1/2, and µ = dz at zero, where z is the usual complex coordinate on C. As in Hirzerbruch, Berger and Jung ([12], Section 2.2), we can associate to this data a function s(z) on C which is elliptic (doubly periodic) with respect to a sublattice ? of index 2 ˜ in ?, namely ? = ˜ Z?1 + 2Z?2. (This leads to a double covering E˜ ? E, and s can be regarded as a rational function on the “doubled” elliptic curve E˜ .) Indeed, we can de?ne s up to a constant by de?ning its divisor to be D = (0) + (?1/2) - (?2) - (?1/2 + ?2) . Then we can make s unique by requiring that ds = dz at zero. We call this s the Jacobi sine. It has the following properties (see [12]): Proposition 4.1. a) s(z) is odd, i.e. s(-z) = -s(z). Around zero, s can be expanded as a power series s(z) = z + a3z 3 + a5z 5 + · · · . b) s(z + ?1) = s(z); s(z + ?2) = -s(z). c) s(z + ?1/2) = a/s(z), a = 0 6 (this follows by looking at the divisor of s(z + ?1/2)). We now show that the construction of s is canonical, i.e. it does not depend on the identi- ?cation E =~ C/?. Proposition 4.2. The de?nition of s only depends on the triple (E, P, µ). Proof. First, we show that the construction of E˜ = C/? is canonical: Let ˜ E =~ C/? 0 be another identi?cation of E. We then have ? 0 = Z? 0 1 + Z? 0 2 , and P is identi?ed with ? 0 1 /2. Since E is also identi?ed with C/?, we get a group map ? : C/? ~ -? C/? 0 . This implies that we have a continuous group map ? : C ~ -? C such that ?(?) = ? 0 . Any such map must be multiplication by a nonzero constant ? ? C. Moreover, we know that ??1/2 = ? 0 1 /2. This implies ??1 = ? 0 1 , and since ? takes ? isomorphically onto ? 0 , it follows that ??2 = ±? 0 2 + m? 0 1 for some integer m. Multiplying this by 2, we get ? · 2?2 = ±2? 0 2 + 2m? 0 1 . This, together with ??1 = ? 0 1 , imply that multiplication by ? descends to a group map C/?˜ ~ -? C/?˜0 . But this precisely means that the construction of E˜ is canonical.10 IOANID ROSU Notice that P can be thought canonically as a point on the “doubled” ellptic curve E˜ . We denote by P1 and P2 the other two points of exact order 2 on E˜ . Then we form the divisor D = (0) + (P ) - (P1) - (P2) . Although the choice of P1 and P2 is noncanonical, the divisor D is canonical, i.e. depends only on P . Let s be an elliptic function on E˜ associated to the divisor D. The choice of s is well-de?ned up to a constant which can be ?xed if we require that ds = p * µ at zero, where p : E˜ ? E is the projection map. Next, we start the construction of S 1 -equivariant elliptic pushforwards. Let f : X ? Y be an equivariant map between compact S 1 -manifolds such that the restrictions f : Xa ? Y a are oriented maps. Then we follow Grojnowski [10] and de?ne the pushforward of f to be a map of sheaves f E ! : E * S1 (X) [f ] ? E * S1 (Y ), where E * S1 (X) [f ] is the sheaf E * S1 (X) twisted by a 1-cocycle to be de?ned later. The main technical ingredient in the construction of the (global i.e. sheafwise) elliptic pushforward f E ! : E * S1 (X) [f ] ? E * S1 (Y ) ,is the (local i.e. stalkwise) elliptic pushforward f E ! : HO* S1 (Xa ) ? HO* S1 (Y a ). In the following subsection, we construct elliptic Thom classes and elliptic pushforwards in HO* S1 (-). The construction is standard, with the only problem that in order to show that something belongs to HO* S1 (-), we need some holomorphicity results on characteristic classes. 4.2. Preliminaries on pushforwards Let p : V ? X be a 2n-dimensional oriented real S 1 -vector bundle over a ?nite S 1 -CW complex X, i.e. a vector bundle with a linear action of S 1 , such that p commutes with the S 1 action. Now, for any space A with an S 1 action, we can de?ne its Borel construction A ×S1 ES 1 , where ES 1 is the universal principal S 1 -bundle. This construction is functorial, so we get a vector bundle VS1 over XS1 . This has a classifying map fV : XS1 ? BSO(2n). If Vuniv is the universal orientable vector bundle over BSO(2n), we also have a map of pairs, also denoted by fV : (DVS1 , SVS1 ) ? (DVuniv, SVuniv). As usual, DV and SV represent the disc and the sphere bundle of V , respectively. But it is known that the pair (DVuniv, SVuniv) is homotopic to (BSO(2n), BSO(2n - 1)). Also, we know that H*BSO(2n) = C[p1, . . . , pn, e]/(e 2 - pn), where pj is the universal j’th Pontrjagin class, and e is the universal Euler class. From the long exact sequence of the pair, it follows that H* (BSO(2n), BSO(2n - 1)) can be regarded as the ideal generated by e in H*BSO(2n). The class e ? H* (DVuniv, SVuniv) is the universal Thom class, which we will denote by funiv. Then the ordinary equivariant Thom class of V is de?ned as the pullback class f * V funiv ? H* S1 (DV, SV ), and we denote it by fS1 (V ). Denote by H** S1 (X) the completion of the module H* S1 (X) with respect to the ideal generated by u in H* (BS 1 ) = C[u]. Consider the power series Q(x) = s(x)/x, where s(x) is the Jacobi sine. Since Q(x) is even, De?nition A.8 gives a class µQ(V )S1 ? H** S1 (X). Then we de?ne a class in H** S1 (DV, SV ) by f E S1 (V ) = µQ(V )S1 · fS1 (V ). One can also say that f E S1 (V ) = s(x1) · · · s(xn), while fS1 (V ) = x1 . . . xn, where x1, . . . , xn are the equivariant Chern roots of V . We call f E S1 (V ) the elliptic equivariant Thom class of V . Also, we de?ne e E S1 (V ), the equivariant elliptic Euler class of V , as the image of f E S1 (V ) via the restriction map H** S1 (DV, SV ) ? H** S1 (X).EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 11 Proposition 4.3. If V ? X is an even dimensional real oriented S 1 -vector bundle, and X is a ?nite S 1 -CW complex, then f E S1 (V ) actually lies in HO* S1 (DV, SV ). Cup product with the elliptic Thom class HO* S1 (X) ? f E S1 (V ) HO* S1 (DV, SV ) , is an isomorphism, the Thom isomorphism in HO-theory. Proof. The di?cult part, namely that µQ(V )S1 is holomorphic, is proved in the Appendix, in Proposition A.6. Consider the usual cup product, which is a map ? : H* S1 (X) ? H* S1 (DV, SV ) ? H* S1 (DV, SV ), and extend it by tensoring with OC,0 over C[u]. We obtain a map ? : HO* S1 (X) ? HO* S1 (DV, SV ) ? HO* S1 (DV, SV ). The equivariant elliptic Thom class of V is f E S1 (V ) = µQ(V )S1 ? fS1 (V ), so we have to show that both these classes are holomorphic. But by Proposition A.6 in the Appendix, µQ(V )S1 ? HO* S1 (X). And the ordinary Thom class fS1 (V ) belongs to H* S1 (DV, SV ), so it also belongs to the larger ring HO* S1 (DV, SV ). Now, cup product with f E S1 (V ) gives an isomorphism because Q(x) = s(x)/x is an invertible power series around zero. Corollary 4.4. If f : X ? Y is an S 1 -equivariant oriented map between compact S 1 - manifolds, then there is an elliptic pushforward f E ! : HO * S1 (X) ? HO * S1 (Y ) , which is a map of HO* S1 (Y )-modules. In the case when Y is a point, f E ! (1) is the S 1 - equivariant elliptic genus of X. Proof. Recall (Dyer [7]) that the ordinary pushforward is de?ned as the composition of three maps, two of which are Thom isomorphisms, and the third is a natural one. The existence of the elliptic pushforward follows therefore from the previous corollary. The proof that f E ! is a map of HO* S1 (Y )-modules is the same as for the ordinary pushforward. The last statement is an easy consequence of the topological Riemann–Roch theorem (see again [7]), and of the de?nition of the equivariant elliptic Thom class. Notice that, if Y is point, HO* S1 (Y ) =~ OC,0, so the S 1 -equivariant elliptic genus of X is holomorphic around zero. Also, if we replace HO* S1 (-) = H* S1 (-) ?C[u] OC,0 by HM* S1 (-) = H* S1 (-) ?C[u] M(C), where M(C) is the ring of global meromorphic functions on C, the same proof as above shows that the S 1 -equivariant elliptic genus of X is meromorphic in C. This proves the following result, which is PROPOSITION B stated in Section 2. Proposition 4.5. The S 1 -equivariant elliptic genus of a compact S 1 -manifold is the Taylor expansion at zero of a function on C which is holomorphic at zero and meromorphic everywhere. 4.3. Construction of f E ! The local construction of elliptic pushforwards is completed. We want now to assemble the pushforwards in a map of sheaves. Let f : X ? Y be a map of compact S 1 -manifolds which commutes with the S 1 -action. We assume that either f is complex oriented or spin oriented, i.e. that the stable normal bundle in the sense of Dyer [7] is complex oriented or spin oriented, respectively. (Grojnowski treats only the complex oriented case, but in order to understand rigidity we also need the spin case.) Let U be an open cover of E adapted to f. Let a, ß ? E be such that Ua n Uß =6 Ø. This implies that at least one point, say ß, is nonspecial, so Xß = XS 1 and Y ß = Y S 1 . We specify12 IOANID ROSU now the orientations of the maps and vector bundles involved. Since Xß = XS 1 , the normal bundle of the embedding Xß ? Xa has a complex structure, where all the weights of the S 1 -action on V are positive. If f is complex oriented, it follows that the restriction maps f a : Xa ? Y a and f ß : Xß ? Y ß are also complex oriented, hence oriented. If f is spin oriented, this means that the stable normal bundle W of f is spin. If H is any subgroup of S 1 , we know that the vector bundle WH ? XH is oriented: If H = S 1 , W splits as a direct sum of WH with a bundle corresponding to the nontrivial irreducible representations of S 1 ; this latter bundle is complex, hence oriented, so the orientation of W induces one on WH . If H = Zn, Lemma 10.3 of Bott and Taubes [4] implies that WH is oriented. In conclusion, both maps f a and f ß are oriented. According to Corollary 4.4, we can de?ne elliptic pushforwards at the level of stalks: (f a ) E ! : HO* S1 (Xa ) ? HO* S1 (Y a ) and (f ß ) E ! : HO* S1 (Xß ) ? HO* S1 (Y ß ). The problem is that pushforwards do not commute with pullbacks, i.e. if i : Xß ? Xa and j : Y ß ? Y a are the inclusions, then it is not true in general that j * (f a ) E ! = (f ß ) E ! i * . However, by twisting the maps with some appropriate Euler classes, the diagram becomes commutative. Denote by e E S1 (Xa /Xß ) the S 1 -equivariant Euler class of the normal bundle to the embedding i, and by e E S1 (Y a /Y ß ) the S 1 -equivariant Euler class of the normal bundle to j. Denote by ? [f ] aß = e E S1 (Xa /Xß ) -1 · (f ß ) * e E S1 (Y a /Y ß ) . A priori ? [f ] aß belongs to the ring HO* S1 (Xß )[ 1 e E S1 (Xa/Xß ) ], but we will see later that we can improve this. Lemma 4.6. In the ring HO* S1 (Xß )[ 1 u , 1 e E S1 (Xa/Xß ) ] we have the following formula j * (f a ) E ! µ a = (f ß ) E ! (i * µ a · ? [f ] aß ) , Proof. From the hypothesis, we know that i * i E ! is an isomorphism, because it is multiplication by the invertible class e E S1 (Xa /Xß ). Also, since u is invertible, the localization theorem implies that i * is an isomorphism. Therefore i E ! is an isomorphism. Start with a class µ a on Xa . Because i E ! is an isomorphism, µ a can be written as i E ! µ ß , where µ ß is a class on Xß . Now look at the two sides of the equation to be proved: 1. The left hand side = j * (f a ) E ! i E ! µ ß = j * j E ! (f ß ) E ! µ ß = (f ß ) E ! µ ß · e E S1 (Y a /Y ß ), because j * j E ! = multiplication by e E (Y a /Y ß ). 2. The right hand side = (f ß ) E ! [i * i E ! µ ß · e E S1 (Xa /Xß ) -1 · (f ß ) * e E S1 (Y a /Y ß )] = (f ß ) E ! [µ ß · (f ß ) * e E S1 (Y a /Y ß )] = (f ß ) E ! µ ß · e E S1 (Y a /Y ß ), where the last equality comes from the fact that (f ß ) E ! is a map of HO* S1 (Y ß )-modules. Let f : X ? Y be a complex or spin oriented S 1 -map. Let U be an open cover adapted to f, and a, ß ? E such that Ua n Uß =6 Ø. We know that a and ß cannot be both special, so assume ß nonspecial. Let U ? Ua n Uß. Since U is adapted, a /? U.EQUIVARIANT ELLIPTIC COHOMOLOGY AND RIGIDITY 13 Proposition 4.7. With these hypotheses, ? [f ] aß belongs to H* S1 (Xß ) ?C[u] OE(U - ß), and the following diagram is commutative: H* S1 (Xa ) ?C[u] OE(U - a) (f a ) E ! ? [f] aß ·i * H* S1 (Y a ) ?C[u] OE(U - a) j * H* S1 (Xß ) ?C[u] OE(U - a) (f ß ) E ! t * ß-a H* S1 (Y ß ) ?C[u] OE(U - a) t * ß-a H* S1 (Xß ) ?C[u] OE(U - ß) (f ß ) E ! H* S1 (Y ß ) ?C[u] OE(U - ß) Proof. Denote by W the normal bundle of the embedding Xß = XS 1 ? Xa . Let us show that, if a /? U, then e E S1 (W) is invertible in H* S1 (Xß ) ?C[u] OE(U - a). Denote by wi the nonequivariant Chern roots of W, and by mi the corresponding rotation numbers of W (see Proposition A.4 in the Appendix). Since Xß = XS 1 , mi = 0. Also, the 6 S 1 -equivariant Euler class of W is given by eS1 (W) = (w1 + m1u) . . . (wr + mru) = m1 . . . mr(u + w1/m1) . . . (u + wr/mr) . But wi are nilpotent, so eS1 (W) is invertible as long as u is invertible. Now a /? U translates to 0 ?/ U - a, which implies that the image of u via the map C[u] ? OE(U - a) is indeed invertible. To deduce now that e E S1 (W), the elliptic S 1 -equivariant Euler class of W, is also invertible, recall that e E S1 (W) and eS1 (W) di?er by a class de?ned using the power series s(x)/x = 1 + a3x 2 + a5x 4 + · · · , which is invertible for U small enough. So ? [f ] aß exists, and by the previous Lemma, the upper part of our diagram is commutative. The lower part is trivially commutative. Now, since i * are essentially the gluing maps in the sheaf F = E * S1 (X), we think of the maps ? [f ] aß · i * as giving the sheaf F twisted by the cocycle ? [f ] aß . Recall from De?nition 3.5 that F was obtained by gluing the sheaves Fa over an adapted open cover (Ua)a?E. De?nition 4.8. The twisted gluing functions f [f ] aß are de?ned as the composition of the following three maps H* S1 (Xa )?C[u] OE(U -a) i *?1 -? H* S1 (Xß )?C[u] OE(U -a) · ? [f] aß -? H* S1 (Xß )?C[u] OE(U - ß) t * ß-a -? H* S1 (Xß ) ?C[u] OE(U - ß). The third map is de?ned as in Remark 3.4. As in the discussion after Remark 3.4, we can show easily that f [f ] aß satisfy the cocycle condition. De?nition 4.9. Let f : X ? Y be an equivariant map of compact S 1 -manifolds, such that it is either complex or spin oriented. We denote by E * S1 (X) [f ] the sheaf obtained by gluing the sheaves Fa de?ned in 3.1, using the twisted gluing functions f [f ] aß . Also, we de?ne the S 1 -equivariant elliptic pushforward of f to be the map of coherent sheaves over E f E ! : E * S1 (X) [f ] ? E * S1 (Y ) which comes from gluing the local elliptic pushforwards (f a ) E ! (as de?ned in 4.4). We call f E ! the Grojnowski pushforward.14 IOANID ROSU The fact that (f a ) E ! glue well comes from the commutativity of the diagram in Proposition 4.7. The Grojnowski pushforward is functorial: see [9] and [10]. 5. Rigidity of the elliptic genus In this section we discuss the rigidity phenomenon in the context of equivariant elliptic cohomology. We start with a discussion about orientations. 5.1. Preliminaries on orientations Let V ? X be an even dimensional spin S 1 -vector bundle over a ?nite S 1 -CW complex X (which means that the S 1 -action preserves the spin structure). Let n ? N. We think of Zn ? S 1 as the ring of n’th roots of unity in C. The invariants of V under the actions of S 1 and Zn are the S 1 -vector bundles V S 1 ? XS 1 and V Zn ? XZn . We have XS 1 ? XZn . Let N be a connected component of XS 1 , and P a connected component of XZn which contains N. From now on we think of V S 1 as a bundle over N, and V Zn as a bundle over P . De?ne the vector bundles V/V S 1 and V Zn /V S 1 over N by V|N = V S 1 ? V/V S 1 ; V Zn |N = V S 1 ? V Zn /V S 1 . The decompositions of these two bundles come from the fact that S 1 acts trivially on the base N, so ?bers decompose into a trivial and nontrivial part. Similarly, the action of Zn on P is trivial, so we get a ?berwise decomposition of V|P by the di?erent representations of Zn: V|P = V Zn ? M 0 |