Riemann–Roch theorems and elliptic genus for virtually smooth schemes

Abstract

For a proper scheme $X$ with a fixed $1$–perfect obstruction theory $E^{\bullet }$, 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.