Sheaves on surfaces and virtual invariants

Abstract

Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which -- in analogy with the classical case of Hilbert schemes of points -- can be used to define intersection numbers, such as virtual Euler characteristics, Verlinde numbers, and Segre numbers. We survey a set of recent conjectures by the authors for these numbers with applications to Vafa-Witten theory, $K$-theoretic S-duality, a rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula, and a virtual Segre-Verlinde correspondence. A key role is played by Mochizuki's formula for descendent Donaldson invariants.

40 pages. Published version

References in corpus (5)

• The Verlinde formula for Higgs bundles
• Invariants of moduli spaces of stable sheaves on ruled surfaces
• The equivariant Verlinde formula on the moduli of Higgs bundles
• Virtual Segre and Verlinde numbers of projective surfaces
• Rank $N$ Vafa-Witten invariants, modularity and blow-up

Cited by in corpus (1)

• $\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings