Riemann-Roch theorems and elliptic genus for virtually smooth Schemes

TL;DR

This paper develops virtual invariants like Euler characteristic, chi y-genus, and elliptic genus for virtually smooth schemes, extending classical theorems and establishing modularity and localization properties.

Contribution

It introduces virtual versions of classical invariants and proves their fundamental properties, including Riemann-Roch theorems and modularity, for schemes with perfect obstruction theories.

Findings

Virtual invariants are deformation invariant.

Virtual chi y-genus is a polynomial.

Virtual elliptic genus satisfies Jacobi modularity.

Abstract

For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.