Algebraic & Geometric Topology Volume 9, issue 4 (2009)

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.

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Keywords

symplectic rational blow-down, surface singularity, symplectic neighborhood

Mathematical Subject Classification

Primary: 57R17

Secondary: 14E15, 14J17

References

Publication

Received: 9 December 2008
Revised: 25 August 2009
Accepted: 9 September 2009
Published: 27 October 2009

Authors

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 Budapest 1053 Hungary	Mathematics Department, Columbia University 2990 Broadway, New York, NY 10027 USA
http://www.renyi.hu/~stipsicz