Riemann-Roch for Singular Varieties

Abstract

The basic tool for a general Riemann-Roch theorem is MacPherson’s graph construction, applied to a complex E. of vector bundles on a scheme Y, exact off a closed subset X. This produces a localized Chern character¹ ch ^y_x (E.) which lives in the bivariant group $A{\left( {X \to Y} \right)_\mathbb{Q}}$ For each class α∈A _* Y, this gives a class

$$ ch_X^Y\left( {E.} \right) \cap \alpha \in {A_ * }{X_\mathbb{Q}} $$

whose image in ${A_ * }{Y_\mathbb{Q}}$ is ${\sum {\left( { - 1} \right)} ^i}ch\left( {{E_i}} \right) \cap \alpha$ The properties needed for Riemann-Roch, in particular the invariance under rational deformation, follow from the bivariant nature of ch ^y_x E.

The general Riemann-Roch theorem constructs homomorphisms

$$ {{\tau }_{x}}:{{K}_{ \circ }}X \to {{A}_{*}}{{X}_{\mathbb{Q}}} $$

covariant for proper morphisms, such that ${\tau _x}\left( {\beta \otimes \alpha } \right) = ch\left( \beta \right) \cap {\tau _x}\left( \alpha \right)$ for $\beta \in {{K}^{ \circ }}X, \alpha \in {{K}_{ \circ }}X$ imbedded in a non-singular variety M, and a coherent sheaf ℐis resolved by a complex of vector bundles E. on M, then

$$ {\tau _x}\left( \mathcal{F} \right) = ch_X^M\left( {E.} \right) \cap Td\left( M \right) $$

where $Td\left( M \right) = td\left( {{T_M}} \right) \cap \left[ M \right]$ Such t_xis constructed for quasi-projective schemes in the second section. The extension to arbitrary algebraic schemes, using Chow’s lemma, is carried out in the last section. As a corollary one has the GRR formula

$$ {f_ * }\left( {ch\left( \alpha \right) \cdot td\left( {{T_x}} \right)} \right) = ch\left( {{f_ * }\alpha } \right) \cdot td\left( {{T_Y}} \right) $$

for f: X→Y proper. X, Y arbitrary non-singular varieties, $\alpha \in {{K}^{ \circ }}X$ In the singular case, there are refinements for f: X→Y a l.c.i. morphism.

Article Footnote

¹ In topology, the Chern character of such a complex lives in; capping with this class determines homomorphisms from. The bivariant class (E.) is an analogue for rational equivalence.