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. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $\omega$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.