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

Lothar Göttsche (International Centre for Theoretical Physics, Trieste, Italy)

Martijn Kool (Mathematical Institute, Utrecht University, Utrecht, The Netherlands)

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.

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Published 29 December 2021