Riemann-Roch theorems and elliptic genus for virtually smooth schemes

By Barbara Fantechi, L. Gottsche. 2007

111 citations recorded by the source corpus.

For a proper scheme X with a fixed 1‐perfect obstruction theory E , we define virtual versions of holomorphic Euler characteristic, 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 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. 14C40; 14C17, 57R20

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e78f5542edc0e2443a89f8b4f29f36e4b64f5f0f

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Refined curve counting on complex surfaces (2012, background): “The description of C ı as zero locus of a section of L H gives it a virtual fundamental class and a virtual tangent bundle (see eg Fantechi and Göttsche [21]).”
Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics (2019, background): “We define the virtual Euler characteristic (36) e(Z) = ∫ [Z] cd(T Z) , see also [10].”