Virtual Segre and Verlinde numbers of projective surfaces

TL;DR

This paper generalizes conjectures relating Segre and Verlinde numbers to moduli spaces of stable sheaves on surfaces with holomorphic 2-forms, deriving universal formulas and verifying them in specific cases, with implications for Donaldson invariants.

Contribution

It extends known conjectures to higher rank sheaves on special surfaces, providing universal formulas and explicit results for Donaldson invariants.

Findings

Derived universal functions expressing invariants in terms of Seiberg-Witten invariants.

Proved certain invariants are topological for general type surfaces.

Verified conjectures in specific examples and obtained explicit formulas for ranks 3 and 4.

Abstract

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 any 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 prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mari\~{n}o-Moore conjecture for higher rank Donaldson invariants. For ranks $3$ and $4$, we obtain explicit expressions for Donaldson invariants in terms of Seiberg-Witten invariants.