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.

arXiv comment40 pages. Published version

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Cited by in corpus (1)

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