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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)


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$.

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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

PII: S 0894-0347(09)00632-8
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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