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

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 $A_{n}$ 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

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.