Gromov-Witten Invariants of Symplectic Sums

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, V ) with a symplectic identification V = V and a complex anti-linear isomorphism between the normal bundles of V and V , we can form the symplectic sum Z = X# V =V 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, V ).Section 1 is a review of the GW invariants for symplectic manifolds and the associated invariants, which we call T W invariants, that count reducible curves.The corresponding relative invariants of a symplectic pair (X, V ) are defined in section 2. The sum formula is stated in a special case in section 3, and in general as Theorem 4.1.The last section presents two applications: 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" (5.3).Related results, involving symplectic sums along contact manifolds, are being developed by Li and Ruan [LR] and by Eliashberg and Hofer.