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Riemann–Roch theorems and elliptic genus for virtually smooth schemes

2010; Mathematical Sciences Publishers; Volume: 14; Issue: 1 Linguagem: Inglês

10.2140/gt.2010.14.83

1465-3060

Barbara Fantechi, Lothar Göttsche,

Homotopy and Cohomology in Algebraic Topology

For a proper scheme X with a fixed 1-perfect obstruction theory, 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.