Quantum cohomology of the Hilbert scheme of points on -resolutions
Authors:
Davesh Maulik and Alexei Oblomkov
Journal:
J. Amer. Math. Soc. 22 (2009), 1055-1091
MSC (2000):
Primary 14N35
DOI:
https://doi.org/10.1090/S0894-0347-09-00632-8
Published electronically:
March 24, 2009
MathSciNet review:
2525779
Full-text PDF Free Access
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 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 Gromov-Witten/Donaldson-Thomas 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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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:
https://doi.org/10.1090/S0894-0347-09-00632-8
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.