Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces

TL;DR

This paper demonstrates a universality principle for descendent integrals over moduli spaces of stable sheaves on K3 surfaces, enabling reduction to simpler integrals and confirming a conjectured higher rank Segre-Verlinde correspondence.

Contribution

It introduces a universality framework for descendent integrals on K3 moduli spaces and proves the higher rank Segre-Verlinde correspondence conjecture.

Findings

Reduction of descendent integrals to punctual Hilbert schemes

Validation of the higher rank Segre-Verlinde conjecture for K3 surfaces

Extension of Markman's monodromy results to universality statements

Abstract

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the $K3$ surface. As an application we establish the higher rank Segre-Verlinde correspondence for $K3$ surfaces as conjectured by G\"ottsche and Kool.