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Surveys in Differential Geometry
Volume 24 (2019)
Sheaves on surfaces and virtual invariants
Pages: 67 – 116
DOI: https://dx.doi.org/10.4310/SDG.2019.v24.n1.a3
Authors
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 $\mathrm{S}$‑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.
Published 29 December 2021