Volume 14, issue 1 (2010)

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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