Quantum cohomology of the Hilbert scheme of points on resolutions
Authors:
Davesh Maulik and Alexei Oblomkov
Journal:
J. Amer. Math. Soc. 22 (2009), 10551091
MSC (2000):
Primary 14N35
Published electronically:
March 24, 2009
MathSciNet review:
2525779
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Abstract: We determine the twopoint invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type singularities. The operators encoding these invariants are expressed in terms of the action of the the affine Lie algebra 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 GromovWitten/DonaldsonThomas theories of . We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of types and .
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 M. Lehn, Ch. Sorger, The cup product of Hilbert schemes for surfaces, Invent. Math. 152 (2003), no. 2, 305329. MR 1974889 (2004a:14004)
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 M. Manetti, Lie description of higher obstructions to deforming submanifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 4, 631659. MR 2394413
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 D. Maulik, A. Oblomkov, DT theory of , 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, GromovWitten theory and NoetherLefschetz theory, arXiv:math/0705.1653.
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 H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), no. 2, 379388. MR 1441880 (98h:14006)
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 A. Okounkov, R. Pandharipande, GromovWitten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517560. MR 2199225 (2007b:14123)
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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:
http://dx.doi.org/10.1090/S0894034709006328
PII:
S 08940347(09)006328
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 DMS0111298 and DMS0701387
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
