Segre and Verlinde formulas for moduli of sheaves on surfaces

This is a report on joint work with Martijn Kool. 

Recently, Marian-Oprea-Pandharipande established a generalization of Lehn’s conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of higher rank. 

Using Mochizuki’s formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg- Witten invariants and intersection numbers on products of Hilbert schemes of points. We use this to  verify our conjectures in examples.