Character Formulas on Cohomology of Deformations of Hilbert Schemes of K3 Surfaces

TL;DR

This paper computes the character formulas for the cohomology of deformed Hilbert schemes of K3 surfaces, revealing invariants of Hodge classes under deformation and providing generating series for their enumeration.

Contribution

It introduces explicit character formulas for the cohomology of deformations of Hilbert schemes of K3 surfaces, advancing understanding of Hodge class invariance.

Findings

Derived a generating series for Hodge classes

Computed graded character formulas for Mumford-Tate group actions

Identified Hodge classes invariant under deformation

Abstract

Let X be a hyperkahler manifold deformation equivalent to a Hilbert scheme of n points on a K3 surface. We compute the graded character formula of the generic Mumford-Tate group representation on the cohomology ring of X, and derive a generating series for deducing the number of canonical Hodge classes on X. The formula indicates the number of Hodge classes on X that remain Hodge under any deformation.