#### Volume 9, issue 4 (2009)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
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