Volume 14, issue 1 (2010)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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, we define virtual versions of holomorphic Euler characteristic, χy–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 χy–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
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
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/