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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 with a fixed –perfect obstruction theory , we define virtual versions of holomorphic Euler characteristic, –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 –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
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Mathematical Subject Classification 2000
Primary: 14C40
Secondary: 14C17, 57R20
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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
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Authors |
| Barbara Fantechi | |
| SISSA Via Beirut 2/4 34151 Trieste Italy |
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| http://people.sissa.it/~fantechi/ | |
| Lothar Göttsche | |
| International Centre for Theoretical
Physics Strada Costiera 11 34151 Trieste Italy |
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| http://users.ictp.it/~gottsche/ | |
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