Riemann-Roch for Singular Varieties

Summary

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 $ch_X^Y \left( {E.} \right)$ which lives in the bivariant group A (X→Y)_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 _ℚ is ∑(−1)ⁱ ch (E _i ) ∩ α. The properties needed for Riemann-Roch, in particular the invariance under rational deformation, follow from the bivariant nature of $ch_X^Y \left( {E.} \right)$.