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.
- Surface singularity
- Symplectic neighborhood
- Symplectic rational blow-down
ASJC Scopus subject areas
- Geometry and Topology