The Symplectic Sum Formula for Gromov-Witten Invariants

Abstract

In the symplectic category there is a `connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z=X#Y in terms of the relative GW invariants of X and Y. Several applications to enumerative geometry are given.

AMS-Latex, 66 pages with 3 eps figures (typos fixed in the introduction)

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• Relative Gromov-Witten Invariants
• On a Proof of a Conjecture of Marino-Vafa on Hodge Integrals
• A degeneration of stable morphisms and relative stable morphisms
• Counting Elliptic Curves in K3 Surfaces
• Computing Hodge integrals with one lambda-class
• Semi-Stable Degeneration of Toric Varieties and Their Hypersurfaces
• Family Gromov-Witten Invariants for Kahler Surfaces
• Symplectic sums and Gromov-Witten invariants
• A Formula of Two-Partition Hodge Integrals
• An index inequality for embedded pseudoholomorphic curves in symplectizations
• A Degeneration formula of GW-invariants