Gromov-Witten Invariants of Symplectic Sums
Abstract
The natural sum operation for symplectic manifolds is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n-2)-submanifold V. Given a similar pair (Y,W) with a symplectic identification V=W and a complex anti-linear isomorphism between the normal bundles of V and W, we can form the symplectic sum Z=X # Y. This note announces a general formula for computing the Gromov-Witten invariants of the sum Z in terms of relative Gromov-Witten invariants of (X,V) and (Y,W). Two applications are presented: a short derivation of the Caporaso-Harris formula [CH], and new proof that the rational enumerative invariants of the rational elliptic surface are given by the ``modular form'' (5.2).
Keywords
Cite
@article{arxiv.math/9806013,
title = {Gromov-Witten Invariants of Symplectic Sums},
author = {Eleny-Nicoleta Ionel and Thomas H. Parker},
journal= {arXiv preprint arXiv:math/9806013},
year = {2007}
}
Comments
AMS-LaTeX, 13 pages; repaginated and 7 typos fixed