Bivariant Riemann Roch Theorems

The goal of this paper is to explain the analogy between certain results in algebraic geometry, namely the Riemann-Roch theorems due to Baum,Fulton, and MacPherson ([BFM2],and [FMac]); and recent results in geometric topology due to Dwyer, Weiss and myself [DWW]. One reason for doing this is that the bivariant viewpoint introduced by Fulton-MacPherson in their memoir [FMac], becomes particularly useful in the topological case. In fact we get that the bivariant topological Riemann-Roch theorem has a converse. Both in the algebraic geometry case and in the geometric topology case we’ll introduce a pair of functors: one of which is covariant and the other contravariant. However, for certain maps between varieties (or topological spaces) we also get transfer (or Gysin) maps for these functors. The Riemann-Roch theorem in both cases gives natural transformations between the above functors and generalized homology/cohomology theories which are compatible with the transfer maps. I. Algebraic Geometry In this paper variety means quasi-projective variety over C. For any variety X , K alg(X) is the Grothendieck group of algebraic C-vector bundles on X . Tensor product makes K alg(X) a ring. For any morphism f : X → Y , there is an induced ring homomorphism f:K alg(Y ) → K 0 alg(X) given by pulling back bundles. This makes K alg a contravariant functor from varieties to commutative rings. Let K top(X) denote the Grothendieck group of topological C-vector bundles on X with the classical topology. Under tensor product K top(X) is a ring, and K top(X) ≃ h (X ;KC). There exists an obvious forgetful natural transformation α:K alg(X) → h (X ;KC). For any variety X, K 0 (X) is the Grothendieck group of coherent sheaves of OX -modules. If f : X → Y is a proper morphism, then the map f∗:K alg 0 (X) → K 0 (Y ) sends [F ] to ∑ i (−1) [(Rf∗F ]. Here R f∗F is the higher direct image 1991 Mathematics Subject Classification. Primary 57R22, 57R55; Secondary 57R57, 57R10, 58G10. Research partially supported by NSF. c ©1997 American Mathematical Society