Journal of Symplectic Geometry

Volume 12 (2014)

Number 3

Smoothings of singularities and symplectic surgery

Pages: 585 – 597



Heesang Park (Department of Mathematics, Konkuk University, Seoul, South Korea)

András I. Stipsicz (Rényi Institute of Mathematics, Budapest, Hungary)


Suppose that $C$ is a connected configuration of two-dimensional symplectic submanifolds in a symplectic 4-manifold with negative definite intersection graph $\Gamma_C$. Let $(S, 0)$ be a normal surface singularity with resolution graph $\Gamma_C$ and suppose that $W_S$ is a smoothing of $(S, 0)$. We show that if we replace an appropriate neighborhood of $C$ with $W_S$, then the resulting 4-manifold admits a symplectic structure. The operation generalizes the rational blow-down operation of Fintushel-Stern, and therefore our result extends Symington’s theorem about symplectic rational blow-downs.

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