Gromov–Witten theory of 𝒜n–resolutions

Maulik, Davesh

doi:10.2140/gt.2009.13.1729

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Gromov–Witten theory of $\mathcal{A}_{n}$–resolutions

Davesh Maulik

Geometry & Topology 13 (2009) 1729–1773

DOI: 10.2140/gt.2009.13.1729

Abstract

We give a complete solution for the reduced Gromov–Witten theory of resolved surface singularities of type $A_{n}$ , for any genus, with arbitrary descendent insertions. We also present a partial evaluation of the $T$ –equivariant relative Gromov–Witten theory of the threefold $A_{n} \times P^{1}$ which, under a nondegeneracy hypothesis, yields a complete solution for the theory. The results given here allow comparison of this theory with the quantum cohomology of the Hilbert scheme of points on the $A_{n}$ surfaces. We discuss generalizations to linear Hodge insertions and to surface resolutions of type $D, E$ . As a corollary, we present a new derivation of the stationary Gromov–Witten theory of $P^{1}$ .

Keywords

Gromov–Witten theory, ADE singularity

Mathematical Subject Classification 2000

Primary: 14N35

References

Publication

Received: 5 March 2008

Revised: 10 December 2008

Accepted: 12 February 2009

Published: 5 March 2009

Proposed: Jim Bryan

Seconded: Yasha Eliashberg, Lothar Goettsche

Authors

Davesh Maulik
Department of Mathematics Columbia University New York, NY 10027 USA

Volume 13, issue 3 (2009)