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

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.

symplectic rational blow-down, surface singularity, symplectic neighborhood
Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 14E15, 14J17
Received: 9 December 2008
Revised: 25 August 2009
Accepted: 9 September 2009
Published: 27 October 2009
David T Gay
Department of Mathematics and Applied Mathematics
University of Cape Town
Private Bag X3
Rondebosch 7701
South Africa
András I Stipsicz
Hungarian Academy of Sciences
Renyi Institute of Mathematics
Reáltanoda utca 13–15
Mathematics Department, Columbia University
2990 Broadway, New York, NY 10027