#### Volume 13, issue 3 (2009)

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

### Davesh Maulik

Geometry & Topology 13 (2009) 1729–1773
##### 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 ${\mathsc{A}}_{n}×{P}^{1}$ 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 ${\mathsc{A}}_{n}$ 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 ${P}^{1}$.