Riemann-Roch for Non-singular Varieties

The Grothendieck-Riemann-Roch theorem (GRR) states that for a proper morphism f: X → Y of non-singular varieties, $$ ch\left( {{f_*}\alpha } \right) \times td\left( {{T_Y}} \right) = {f_*}\left( {ch\left( \alpha \right) \times td\left( {{T_X}} \right)} \right) $$ for all α in the Grothendieck group of vector bundles, or of coherent sheaves, on X. When Y is a point, one recovers Hirzebruch’s formula (HRR) for the Euler characteristic of a vector bundle E on X: $$ {\sum {\left( { - 1} \right)} ^i}\dim {H^i}\left( {X,E} \right) = \int\limits_X {ch} \left( E \right) \cdot td\left( {{T_X}} \right) $$ The aim of this chapter is to show how the geometry of the deformation to the normal cone leads to a simple proof of GRR when f is a closed imbedding. The same proof gives the corresponding theorem without denominators, which in turn yields a simple proof of the formula for blowing up Chern classes. The reader of this chapter is assumed to have some familiarity with the cohomology of coherent sheaves, although the necessary facts are reviewed in the first section. In addition, the proof of GRR when f is a projection is only sketched briefly. The first nine sections of the article of Borel and Serre (1) are recommended for a detailed discussion of these points. Although the theorem is stated here for arbitrary non-singular varieties, the proof in this chapter makes an additional assumption of projectivity. The general case will be considered, together with singular varieties, in Chap. 18.