Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants

TL;DR

This paper proposes conjectural blowup formulas for Segre and Verlinde numbers on moduli spaces of sheaves on surfaces, leading to new conjectures for Donaldson invariants in various ranks and settings.

Contribution

It introduces conjectural formulas for blowup behavior of Segre and Verlinde numbers, extending to Donaldson invariants of surfaces in arbitrary rank and K-theoretic variants.

Findings

Conjectural blowup formulas for Segre and Verlinde numbers.

Proposed formulas for Donaldson invariants in arbitrary rank.

Extensions to K-theoretic Donaldson invariants and invariants with fundamental matter.

Abstract

We formulate conjectural blowup formulas for Segre and Verlinde numbers on moduli spaces of sheaves on projective surfaces $S$ with $p_g(S)>0$ and $b_1(S)=0$. As applications we give a give a conjectural formula for the Donaldson invariants of $S$ in arbitrary rank, as well as for the $K$-theoretic Donaldson invariants, and some Donaldson invariants with fundamental matters.