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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Quantum cohomology of the Hilbert scheme of points on $\mathcal {A}_n$-resolutions
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by Davesh Maulik and Alexei Oblomkov PDF
J. Amer. Math. Soc. 22 (2009), 1055-1091 Request permission

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
  • Email: dmaulik@math.mit.edu
  • Alexei Oblomkov
  • Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
  • Email: oblomkov@math.princeton.edu
  • 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
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 22 (2009), 1055-1091
  • MSC (2000): Primary 14N35
  • DOI: https://doi.org/10.1090/S0894-0347-09-00632-8
  • MathSciNet review: 2525779