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

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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.