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Quantum cohomology of the Hilbert scheme of points on $ \mathcal{A}_n$-resolutions

Authors: Davesh Maulik and Alexei Oblomkov
Journal: J. Amer. Math. Soc. 22 (2009), 1055-1091
MSC (2000): Primary 14N35
Published electronically: March 24, 2009
MathSciNet review: 2525779
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Abstract: We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type $ A_{n}$ singularities. The operators encoding these invariants are expressed in terms of the action of the the affine Lie algebra $ \widehat{\mathfrak{gl}}(n+1)$ on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of $ A_{n}\times\mathbf{P}^1$. We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of types $ D$ and $ E$.

References [Enhancements On Off] (What's this?)

  • [B] A. Borel, Algebraic D-modules, Academic Press, Orlando, Florida, 1987. MR 0882000 (89g:32014)
  • [BKL] J. Bryan, S. Katz, N.C. Leung, Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds, J. Algebraic Geom. 10 (3) (2001), 549-568. MR 1832332 (2002j:14047)
  • [BP] J. Bryan, R. Pandharipande, The local Gromov-Witten theory of curves, JAMS 21 (2008), 101-136. MR 2350052 (2008h:14057)
  • [GP] T. Graber, R. Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487-518. MR 1666787 (2000h:14005)
  • [Gr] I. Grojnowski, Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), no. 2, 275-291. MR 1386846 (97f:14041)
  • [K] T. Kato, Perturbation theory for linear operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. MR 1335452 (96a:47025)
  • [KL] Young-Hoon Kiem, Jun Li, Gromov-Witten invariants of varieties with holomorphic 2-forms, arXiv:math/0707.2986.
  • [KM] M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 164 (1994), no. 3, 525-562. MR 1291244 (95i:14049)
  • [LS] M. Lehn, Ch. Sorger, The cup product of Hilbert schemes for $ K3$ surfaces, Invent. Math. 152 (2003), no. 2, 305-329. MR 1974889 (2004a:14004)
  • [LQW] W. Li, Z. Qin, W. Wang, The cohomology rings of Hilbert schemes via Jack polynomials, CRM Proceedings and Lecture Notes 38 (2004), 249-258. MR 2096149 (2005k:14009)
  • [Man] M. Manetti, Lie description of higher obstructions to deforming submanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 4, 631-659. MR 2394413
  • [M] D. Maulik, GW theory of $ \mathcal{A}_n$ resolutions, to appear in Geometry and Topology, arXiv:math/0802.2681.
  • [MO] D. Maulik, A. Oblomkov, DT theory of $ \mathcal{A}_n\times\mathbf{P}^1$, to appear in Compos. Math., arXiv:math/0802.2739.
  • [MOOP] D. Maulik, A. Oblomkov, A. Okounkov, R. Pandharipande, GW/DT correspondence for toric threefolds, arXiv:math/0809.3976.
  • [MP] D. Maulik, R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory, arXiv:math/0705.1653.
  • [N1] H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), no. 2, 379-388. MR 1441880 (98h:14006)
  • [N2] H. Nakajima, Jack polynomials and Hilbert schemes of points on surfaces, 1996, arXiv:math/9610021.
  • [N3] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, American Mathematical Society, Providence, RI, 1999. MR 1711344 (2001b:14007)
  • [OP1] A. Okounkov, R. Pandharipande, Quantum cohomology of the Hilbert scheme of points in the plane, arXiv:math/0411210.
  • [OP2] A. Okounkov, R. Pandharipande, The local Donaldson-Thomas theory of curves, arXiv:math/0512573.
  • [OP3] A. Okounkov, R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517-560. MR 2199225 (2007b:14123)
  • [QW] Z. Qin, W. Wang, Hilbert schemes of points on the minimal resolution and soliton equations, Contemp. Math. 442 (2007), 435-462. MR 2372578 (2009b:14008)
  • [R] Z. Ran, Semiregularity, obstructions and deformations of Hodge classes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 809-820. MR 1760539 (2001g:14014)

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Additional Information

Davesh Maulik
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Alexei Oblomkov
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Keywords: Hilbert scheme of points, quantum cohomology
Received by editor(s): March 5, 2008
Published electronically: March 24, 2009
Additional Notes: The first author was partially supported by an NSF Graduate Fellowship and a Clay Research Fellowship
The second author was partially supported by NSF grants DMS-0111298 and DMS-0701387
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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