Riemann--Roch theorem for operations in cohomology of algebraic varieties

Abstract

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 )c oincides with the 2nth component of the characteristic class ch(E )t d(X) under the identification H 2n (X(C), Q )= Q ,w heren =d imX ,c h(E) is the Chern character of E ,a nd 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 ,w here td(L) is obtained by substituting z = c1(L )i n the series (1) td(z )= z