#### Volume 14, issue 1 (2010)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Riemann–Roch theorems and elliptic genus for virtually smooth schemes

### Barbara Fantechi and Lothar Göttsche

Geometry & Topology 14 (2010) 83–115
##### 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.

##### Keywords
Riemann–Roch theorems, virtual fundamental class, genus
##### Mathematical Subject Classification 2000
Primary: 14C40
Secondary: 14C17, 57R20
##### 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
 Barbara Fantechi SISSA Via Beirut 2/4 34151 Trieste Italy http://people.sissa.it/~fantechi/ Lothar Göttsche International Centre for Theoretical Physics Strada Costiera 11 34151 Trieste Italy http://users.ictp.it/~gottsche/