Symplectic divisorial capping in dimension $4$

We investigate the notion of symplectic divisorial compactification for symplectic $4$-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. Our main classification result is that if the symplectic form of a symplectic divisor is exact on the boundary of its plumbing, then the symplectic divisor admits either a concave or convex neighborhood after a symplectic deformation that keeps the divisor symplectic.

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Received 3 May 2018

Accepted 10 August 2018

Published 17 January 2020