Hostname: page-component-6898cfcb55-th4rj Total loading time: 0 Render date: 2026-11-11T12:13:46.757Z Has data issue: false hasContentIssue false
  • English
  • Français

Donaldson–Thomas theory of 𝒜n×P1

Part of:Projective and enumerative geometry

Published online by Cambridge University Press:  18 August 2009

Davesh Maulik and
Alexei Oblomkov
Davesh Maulik
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA (email: dmaulik@cpw.math.columbia.edu)
Alexei Oblomkov
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA (email: oblomkov@math.princeton.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We study the relative Donaldson–Thomas theory of 𝒜n×P1, where 𝒜n is the surface resolution of type An singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra $\glh $ on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov–Witten theory of 𝒜n×P1 and the quantum cohomology of the Hilbert scheme of points on 𝒜n.

Keywords

Donaldson–Thomas theoryloop algebrasresolution of quotient surface singularities

MSC classification

Secondary:14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009, pp. 1249 - 1276
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Bryan, J. and Pandharipande, R., The local Gromov–Witten theory of curves, J. Amer. Math. Soc. 21 (2008), 101–136.Google Scholar
[2]Graber, T. and Pandharipande, R., Localization of virtual classes, Invent. Math. 135 (1999), 487–518.CrossRefGoogle Scholar
[3]Grojnowski, I., Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275–291.CrossRefGoogle Scholar
[4]Li, J., Liu, C.-C. M., Liu, K. and Zhou, J., A mathematical theory of the topological vertex, Preprint (2004), math/0408426.Google Scholar
[5]Maulik, D., Gromov–Witten theory of 𝒜n resolutions, Geom. Topol. 13 (2009), 1729–1773.CrossRefGoogle Scholar
[6]Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov–Witten theory and Donaldson–Thomas theory I, Compositio Math. 142 (2006), 1263–1285.CrossRefGoogle Scholar
[7]Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov–Witten theory and Donaldson–Thomas theory II, Compositio Math. 142 (2006), 1286–1304.CrossRefGoogle Scholar
[8]Maulik, D. and Oblomkov, A., Quantum cohomology of the Hilbert scheme of points on𝒜n-resolutions, J. Amer. Math. Soc. 22 (2009), 1055–1091.CrossRefGoogle Scholar
[9]Maulik, D., Oblomkov, A., Okounkov, A. and Pandharipande, R., GW/DT correspondence for toric threefolds, Preprint (2008), math/0809.3976.Google Scholar
[10]Maulik, D. and Pandharipande, R., A topological view of Gromov–Witten theory, Topology 45 (2006), 887–918.CrossRefGoogle Scholar
[11]Nakajima, H., Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), 379–388.CrossRefGoogle Scholar
[12]Okounkov, A. and Pandharipande, R., Quantum cohomology of the Hilbert scheme of points in the plane, Preprint (2004), math/0411210.Google Scholar
[13]Okounkov, A. and Pandharipande, R., The local Donaldson–Thomas theory of curves, Preprint (2005), math/0512573.Google Scholar
[14]Pandharipande, R. and Thomas, R., Curve counting via stable pairs in the derived category, Preprint (2007), math/0707.2348.Google Scholar
[15]Qin, Z. and Wang, W., Hilbert schemes of points on the minimal resolution and soliton equations, Contemp. Math. 442 (2007), 435–462.CrossRefGoogle Scholar
[16]Wu, B., The moduli stack of stable relative ideal sheaves, Preprint (2007), math/0701074.Google Scholar