Virtual Segre and Verlinde numbers of projective surfaces

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 Mariño-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.

Published version. 38 pages

References in corpus (5)

• Virtual Segre and Verlinde numbers of projective surfaces
• Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces
• Vertical Vafa-Witten invariants
• $\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
• Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants

Cited by in corpus (9)

• The virtual K-theory of Quot schemes of surfaces
• Virtual Segre and Verlinde numbers of projective surfaces
• Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces
• Sheaves on surfaces and virtual invariants
• $\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants
• Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points
• Equivariant Segre and Verlinde invariants for Quot schemes