Tunisian Journal of Mathematics

Degeneracy loci, virtual cycles and nested Hilbert schemes, I

Amin Gholampour and Richard P. Thomas

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Given a map of vector bundles on a smooth variety, consider the deepest degeneracy locus where its rank is smallest. We show it carries a natural perfect obstruction theory whose virtual cycle can be calculated by the Thom–Porteous formula.

We show nested Hilbert schemes of points on surfaces can be expressed as degeneracy loci. We show how to modify the resulting obstruction theories to recover the virtual cycles of Vafa–Witten and reduced local DT theories. The result computes some Vafa–Witten invariants in terms of Carlsson–Okounkov operators. This proves and extends a conjecture of Gholampour, Sheshmani, and Yau and generalises a vanishing result of Carlsson and Okounkov.

Article information

Tunisian J. Math., Volume 2, Number 3 (2020), 633-665.

Received: 11 February 2019
Accepted: 20 June 2019
First available in Project Euclid: 13 December 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx] 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14C05: Parametrization (Chow and Hilbert schemes) 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

Hilbert scheme degeneracy locus Thom–Porteous formula local Donaldson–Thomas theory Vafa–Witten invariants


Gholampour, Amin; Thomas, Richard P. Degeneracy loci, virtual cycles and nested Hilbert schemes, I. Tunisian J. Math. 2 (2020), no. 3, 633--665. doi:10.2140/tunis.2020.2.633. https://projecteuclid.org/euclid.tunis/1576206291

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