Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces

TL;DR

This paper provides explicit generators for the cohomology ring of moduli spaces of stable sheaves on symplectic surfaces, including K3 and Abelian surfaces, and relates these to universal sheaves and known results for Higgs bundles.

Contribution

It expresses the diagonal class in terms of universal sheaf Chern classes and identifies generators of the cohomology ring, extending known results to new classes of moduli spaces.

Findings

Generators of the cohomology ring are explicitly constructed.

In the K3 case, the generators are free of relations in stable cohomology.

Recovers known results for Higgs bundle moduli spaces.

Abstract

Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n subschemes, this set of generators is sufficiently small in the sense that there aren't any relations among them in the stable cohomology ring. When S is the cotangent bundle of a Riemann surface, we recover the result of T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces of Higgs bundles is generated by the universal classes.