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G&T Monographs |
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Riemann–Roch theorems and elliptic genus for
virtually smooth schemes
Barbara Fantechi and Lothar Göttsche
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Geometry & Topology 14 (2010)
83–115
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Abstract
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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.
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Keywords
Riemann–Roch theorems, virtual
fundamental class, genus
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Mathematical Subject Classification
Primary: 14C40
Secondary: 14C17, 57R20
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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
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