Geometry & Topology
- Geom. Topol.
- Volume 14, Number 1 (2010), 83-115.
Riemann–Roch theorems and elliptic genus for virtually smooth schemes
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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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. https://projecteuclid.org/euclid.gt/1513732172