Geometry & Topology

Riemann–Roch theorems and elliptic genus for virtually smooth schemes

Barbara Fantechi and Lothar Göttsche

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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, 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.

Article information

Geom. Topol., Volume 14, Number 1 (2010), 83-115.

Received: 7 February 2008
Revised: 15 May 2009
Accepted: 7 September 2009
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 14C40: Riemann-Roch theorems [See also 19E20, 19L10]
Secondary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 57R20: Characteristic classes and numbers

Riemann–Roch theorems virtual fundamental class genus


Fantechi, Barbara; Göttsche, Lothar. Riemann–Roch theorems and elliptic genus for virtually smooth schemes. Geom. Topol. 14 (2010), no. 1, 83--115. doi:10.2140/gt.2010.14.83.

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