Symplectic Gluing and Family Gromov-Witten Invariants

Details Symplectic Gluing and Family Gromov-Witten Invariants Symplectic Gluing and Family Gromov-Witten Invariants

2005-03-09

arxiv.org/abs/math/0503176v2

Mathematics

Algebraic Geometry

26 pages, 7 figures. Survey paper to appear in the Proceedings of the Fields-McMaster Conference on the Geometry and Topology

Scientific paper

This article describes the use of symplectic cut-and-paste methods to compute Gromov-Witten invariants. Our focus is on recent advances extending these methods to Kahler surfaces with geometric genus p_g>0, for which the usual GW invariants vanish for most homology classes. This involves extending the Splitting Formula and the Symplectic Sum Formula to the family GW invariants introduced by the first author. We present applications to the invariants of elliptic surfaces and to the Yau-Zaslow Conjecture. In both cases the results agree with the conjectures of algebraic geometers and yield a proof, to appear in [LL1], of previously unproved cases of the Yau-Zaslow Conjecture.

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