Vol. 2, No. 3, 2020

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Degeneracy loci, virtual cycles and nested Hilbert schemes, I

Amin Gholampour and Richard P. Thomas

Vol. 2 (2020), No. 3, 633–665

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.

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Hilbert scheme, degeneracy locus, Thom–Porteous formula, local Donaldson–Thomas theory, Vafa–Witten invariants
Mathematical Subject Classification 2010
Primary: 14D20, 14J60, 14N35
Secondary: 14C05, 57R57
Received: 11 February 2019
Accepted: 20 June 2019
Published: 9 October 2019
Amin Gholampour
Department of Mathematics
University of Maryland
United States
Richard P. Thomas
Department of Mathematics
Imperial College
United Kingdom