Available in electronic format
Available in print format		 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN: 1088-6834(e) ISSN: 0894-0347(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

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

Author(s): Davesh Maulik; Alexei Oblomkov
Journal: J. Amer. Math. Soc. 22 (2009), 1055-1091.
MSC (2000): Primary 14N35
Posted: March 24, 2009
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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:

[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)

Similar Articles:

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 14N35

Retrieve articles in all Journals with MSC (2000): 14N35

Additional Information:

Davesh Maulik
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: dmaulik@math.mit.edu

Alexei Oblomkov
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: oblomkov@math.princeton.edu

DOI: 10.1090/S0894-0347-09-00632-8
PII: S 0894-0347(09)00632-8
Keywords: Hilbert scheme of points, quantum cohomology
Received by editor(s): March 5, 2008
Posted: 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
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google