## Quantum cohomology of the Hilbert scheme of points on $\mathcal {A}_n$-resolutions

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- by Davesh Maulik and Alexei Oblomkov
- J. Amer. Math. Soc.
**22**(2009), 1055-1091 - DOI: https://doi.org/10.1090/S0894-0347-09-00632-8
- Published electronically: March 24, 2009
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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 $E$.## References

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## Bibliographic 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