Riemann–roch Theorem for Operations in Cohomology of Algebraic Varieties

The Riemann–Roch theorem for multiplicative operations in oriented cohomology theories for algebraic varieties is proved and an explicit formula for the corresponding Todd classes is given. The formula obtained can also be applied in the topological situation, and the theorem can be regarded as a change-of-variables formula for the integration of cohomology classes. The classical Riemann–Roch theorem [1], stated and proved by Hirzebruch, calculates the Euler characteristic of a vector bundle on a smooth projective algebraic variety X/C in terms of its rank and Chern classes. This theorem states that χ(E) coincides with the 2nth component of the characteristic class ch(E) td(X) under the identification H(X(C),Q) = Q, where n = dimX, ch(E) is the Chern character of E, and td(X) is the Todd class of the tangent bundle TX . By the splitting principle, the calculation of the Todd classes reduces to the case of a line bundle L, where td(L) is obtained by substituting z = c1(L) in the series (1) td(z) = z 1− e−z . Grothendieck’s generalization of the Riemann–Roch theorem [2] deals with the ring K0 of virtual bundles and calculates the rank and rational Chern classes of a certain virtual bundle, namely, of the direct image of E under a proper mapping p : X → Y (the Hirzebruch theorem is obtained if Y = pt, and χ(E) is regarded as the rank of the direct image in K0). After the functor K0 was carried over to topology and was extended to a cohomology theory, the corresponding version of the Riemann–Roch theorem arose [3]. A more general topological Riemann–Roch theorem appeared in the paper [4] by E. Dyer, where he mentioned the folklore origin of the theorem and the fact that the theorem was known to Adams, Atiyah, and Hirzebruch. In that theorem, the Chern character is replaced by an arbitrary multiplicative operation between arbitrary cohomology theories (the varietiesX and Y were assumed to be oriented in each of the two cohomology theories). When motivic versions of topological theories appeared [5, 6, 7] and transfers for oriented theories were constructed [8] (see [9, 10] for a complete account), a natural problem arose: to generalize the Riemann–Roch theorem to the motivic situation. Such a theorem was stated, and two proofs of it were outlined in [8]. In the present paper, we present the first of these proofs. This proof involves a direct calculation based on the invariance of the residue. The corresponding Todd classes are defined by a simple formula (see Definition 2.2.1) generalizing (1). We show that the same formula can be used in the topological setting. Furthermore, the Riemann–Roch 2000 Mathematics Subject Classification. Primary 14F25, 14F42, 14F43, 14F99.