Volume 9, issue 4 (2009)

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Symplectic surgeries and normal surface singularities

David T Gay and András I Stipsicz

Algebraic & Geometric Topology 9 (2009) 2203–2223
Abstract

We show that every negative definite configuration of symplectic surfaces in a symplectic $4$–manifold has a strongly symplectically convex neighborhood. We use this to show that if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic $4$–manifolds.

Keywords
symplectic rational blow-down, surface singularity, symplectic neighborhood
Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 14E15, 14J17