Tunisian Journal of Mathematics

Degeneracy loci, virtual cycles and nested Hilbert schemes, I

Amin Gholampour and Richard P. Thomas

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/tunis.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.tunis/1576206291

Digital Object Identifier
doi:10.2140/tunis.2020.2.633

Mathematical Reviews number (MathSciNet)
MR4041285

Subjects
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]

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

Citation

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


Export citation

References

  • K. Behrend and B. Fantechi, “The intrinsic normal cone”, Invent. Math. 128:1 (1997), 45–88.
  • E. Carlsson and A. Okounkov, “Exts and vertex operators”, Duke Math. J. 161:9 (2012), 1797–1815.
  • M. Dürr, A. Kabanov, and C. Okonek, “Poincaré invariants”, Topology 46:3 (2007), 225–294.
  • D. Eisenbud, Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer, 1995.
  • W. Fulton, Intersection theory, Ergebnisse der Mathematik $($3$)$ 2, Springer, 1984.
  • A. Gholampour and R. P. Thomas, “Degeneracy loci, virtual cycles and nested Hilbert schemes, II”, preprint, 2019.
  • A. Gholampour, A. Sheshmani, and S.-T. Yau, “Localized Donaldson–Thomas theory of surfaces”, preprint, 2017.
  • A. Gholampour, A. Sheshmani, and S.-T. Yau, “Nested Hilbert schemes on surfaces: virtual fundamental class”, preprint, 2017.
  • M. Kool and R. Thomas, “Reduced classes and curve counting on surfaces, II: Calculations”, Algebr. Geom. 1:3 (2014), 384–399.
  • A. Kresch, “Cycle groups for Artin stacks”, Invent. Math. 138:3 (1999), 495–536.
  • L. Manivel, “Chern classes of tensor products”, Int. J. Math. 27:10 (2016), art. id. 1650079.
  • D. Maulik and A. Okounkov, “Nested Hilbert schemes and symmetric functions”, unpublished.
  • A. Negu\commaaccentt, “Moduli of flags of sheaves and their $K$-theory”, Algebr. Geom. 2:1 (2015), 19–43.
  • A. Negu\commaaccentt, “Shuffle algebras associated to surfaces”, Selecta Math. $($N.S.$)$ 25:3 (2019), art. id. 36.
  • A. Sheshmani and S.-T. Yau, “Flag of sheaves on surfaces, I: Virtual fundamental classes”, in preparation.
  • Y. Tanaka and R. P. Thomas, “Vafa–Witten invariants for projective surfaces, I: Stable case”, preprint, 2017. to appear in J. Algebraic Geom.