Riemann–Roch theorems and elliptic genus for virtually smooth schemes

Fantechi, Barbara¹ ; Göttsche, Lothar²

¹ SISSA, Via Beirut 2/4, 34151 Trieste, Italy
² International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy

Geometry & topology, Tome 14 (2010) no. 1, pp. 83-115.

Résumé

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.

Détail

DOI : 10.2140/gt.2010.14.83

Keywords: Riemann–Roch theorems, virtual fundamental class, genus

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