Geometry & Topology 14 (2010) 83–115

DOI: 10.2140/gt.2010.14.83

Abstract

For a proper scheme X with a fixed 1–perfect obstruction theory E^∙, we define virtual versions of holomorphic Euler characteristic, χ_−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 χ_−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.

Keywords

Riemann–Roch theorems, virtual fundamental class, genus

Mathematical Subject Classification

Primary: 14C40

Secondary: 14C17, 57R20

References

Publication

Received: 7 February 2008
Revised: 15 May 2009
Accepted: 7 September 2009
Preview posted: 10 October 2009
Published: 2 January 2010
Proposed: Jim Bryan
Seconded: Richard Thomas, Peter Ozsváth

Authors