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Degeneracy loci, virtual cycles and nested Hilbert schemes II

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

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, [email protected]
Richard P. Thomas
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, [email protected]
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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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References

Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.CrossRefGoogle Scholar
Carlsson, E. and Okounkov, A., Exts and vertex operators, Duke Math. J. 161 (2012), 1797–1815.CrossRefGoogle Scholar
Chang, H. L. and Kiem, Y. H., Poincaré invariants are Seiberg–Witten invariants, Geom. Topol. 17 (2013), 1149–1163.10.2140/gt.2013.17.1149CrossRefGoogle Scholar
Dürr, M., Kabanov, M. A. and Okonek, C., Poincaré invariants, Topology 46 (2007), 225–294.CrossRefGoogle Scholar
Eisenbud, D., Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, 1994).Google Scholar
Ellingsrud, G., Göttsche, L. and Lehn, M., On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001), 81–100.Google Scholar
Fulton, W., Intersection theory (Springer, 1998).CrossRefGoogle Scholar
Gholampour, A., Sheshmani, A. and Yau, S.-T., Nested Hilbert schemes on surfaces: virtual fundamental class, Adv. Math. 365 (2020), doi:10.1016/j.aim.2020.107046.CrossRefGoogle Scholar
Gholampour, A., Sheshmani, A. and Yau, S.-T., Localized Donaldson–Thomas theory of surfaces, Amer. J. Math. 142 (2020), 405–442.CrossRefGoogle Scholar
Gholampour, A. and Thomas, R. P., Degeneracy loci, virtual cycles and nested Hilbert schemes I, Tunis. J. Math. 2 (2020), 633–665.CrossRefGoogle Scholar
Göttsche, L. and Kool, M., Virtual refinements of the Vafa–Witten formula, Comm. Math. Phys. 376 (2020), 1–49.CrossRefGoogle Scholar
Göttsche, L., Nakajima, H. and Yoshioka, K., Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008), 343–390.CrossRefGoogle Scholar
Jouanolou, J. P., Une suite exacte de Mayer–Vietoris en K-théorie algébrique, in Higher K-theories (Springer, 1973), 293–316.10.1007/BFb0067063CrossRefGoogle Scholar
Kollár, J., Projectivity of complete moduli, J. Differential Geom. 32 (1990), 235–268.CrossRefGoogle Scholar
Kool, M., Stable pair invariants of surfaces and Seiberg–Witten invariants, Q. J. Math. 67 (2016), 365–386.CrossRefGoogle Scholar
Kool, M. and Thomas, R. P., Reduced classes and curve counting on surfaces I: theory, Algebr. Geom. 1 (2014), 334–383.CrossRefGoogle Scholar
Kool, M. and Thomas, R. P., Reduced classes and curve counting on surfaces II: calculations, Algebr. Geom. 1 (2014), 384–399.CrossRefGoogle Scholar
Kresch, A., Cycle groups for Artin stacks, Invent. Math. 138 (1999), 495–536.CrossRefGoogle Scholar
Laarakker, T., Monopole contributions to refined Vafa–Witten invariants, Geom. Topol., to appear. Preprint (2018), arXiv:1810.00385.Google Scholar
Laarakker, T., Vertical Vafa–Witten invariants, Preprint (2020), arXiv:1906.01264.Google Scholar
Manivel, L., Chern classes of tensor products, Internat. J. Math. 27 (2016), 1650079.CrossRefGoogle Scholar
Siebert, B., Virtual fundamental classes, global normal cones and Fulton's canonical classes, in Frobenius manifolds (Vieweg, 2004), 341–358.CrossRefGoogle Scholar
Tanaka, Y. and Thomas, R. P., Vafa–Witten invariants for projective surfaces II: semistable case, Pure Appl. Math. Q. 13 (2017), 517–562, volume in honour of the 60th birthday of Simon Donaldson.CrossRefGoogle Scholar
Tanaka, Y. and Thomas, R. P., Vafa–Witten invariants for projective surfaces I: stable case, J. Algebr. Geom. 29 (2020), 603–668.CrossRefGoogle Scholar
Thomas, R. P., Equivariant K-theory and refined Vafa–Witten invariants, Comm. Math. Phys. 378 (2020), 1451–1500.CrossRefGoogle Scholar