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Degeneracy loci, virtual cycles and nested Hilbert schemes II

Part of: Cycles and subschemes Projective and enumerative geometry Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  01 October 2020

Amin Gholampour  and
Richard P. Thomas
Amin Gholampour
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD20742, USAamingh@math.umd.edu
Richard P. Thomas
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UKrichard.thomas@imperial.ac.uk
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Abstract

We express nested Hilbert schemes of points and curves on a smooth projective surface as ‘virtual resolutions’ of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa–Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom–Porteous-like Chern class formulae.

Keywords

Hilbert schemedegeneracy locusThom–Porteous formulalocal Donaldson–Thomas theoryVafa–Witten invariants

MSC classification

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 14C05: Parametrization (Chow and Hilbert schemes)
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 8 , August 2020 , pp. 1623 - 1663
Copyright
Copyright © The Author(s) 2020

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