Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on K3 surfaces

We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on moduli spaces of sheaves on K3 surfaces.

Canonical holomorphic sections of determinant line bundles

We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

The Determinant Line Bundle for Fredholm Operators: Construction, Properties, and Classification

We provide a thorough construction of a system of compatible determinant line bundles over spaces of Fredholm operators, fully verify that this system satisfies a number of important properties, and include explicit formulas for all relevant isomorphisms between these line bundles. We also completely describe all possible systems of compatible determinant line bundles and compare the conventions and approaches used elsewhere in the literature.

On the S-fundamental group scheme. II

The S-fundamental group scheme is the group scheme corresponding to the Tannaka category of numerically flat vector bundles. We use determinant line bundles to prove that the S-fundamental group of a product of two complete varieties is a product of their S-fundamental groups as conjectured by Mehta and the author. We also compute the abelian part of the S-fundamental group scheme and the S-fundamental group scheme of an abelian variety or a variety with trivial étale fundamental group.