Sympletic normal connect sum

Abstract

Since the publication in 1985 of Gromov’s paper [G1] on pseudo-holomorphic curves in symplectic manifolds there has been an increased interest in symplectic manifolds and symplectic topology. In particular, compact symplectic manifolds have become a focus of much study. In 1977 Thurston [T] gave an example of a compact symplectic manifold with first Betti number three, showing that not all compact symplectic manifolds admit a Kahler structure. However, the difference between the family of compact symplectic manifolds and compact Kahler manifolds remains unclear. In fact there are essentially only two general procedures for constructing compact symplectic manifolds: the symplectic fibration construction, originally due to Thurston, and blowing up along symplectic submanifolds, introduced by Gromov [G2]. Recently R. Gompf has introduced a new construction. He considers two symplectic 4-manifolds each containing the compact surface Σ symplectically embedded with trivial normal bundle. By the symplectic neighborhood theorem a tubular neighborhood of Σ in each 4-manifold is symplectomorphic to Σ×D2 equipped with the product symplectic structure. It follows then that the complements of the tubular neighborhoods of Σ in the symplectic 4-manifolds can be symplectically glued together along tubular shell neighborhoods of Σ by the map Id × φ where φ is an area preserving map of the annulus which interchanges the boundaries. Gompf proceeded by using this construction to show that a compact simply-connected 4-manifold not admitting any complex structure, which he constructed with T. Mrowka [GM], admits a symplectic structure. He thus produced the first example of a compact simply-connected symplectic 4-manifold not admitting any Kahler structure. In this paper we introduce a construction of four dimensional symplectic manifolds, that we call symplectic normal connect sum which generalizes Gompf’s construction. Our procedure constructs a new symplectic 4manifold X = X−1#ΨX1 from pairs (Xi,Σi), i = −1, 1, where the Xi are symplectic 4-manifolds and the Σi are compact embedded symplectic surfaces of genus g and of self-intersection n (for i = 1) and −n (for i = −1), n ≥ 0. We symplectically glue the complements of tubular neighborhoods of Σ−1 in X−1 and Σ1 in X1 along tubular shell neighborhoods of Σ−1 and