Symplectic divisorial capping in dimension $4$

Abstract

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 ω-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.

Strong symplectic fillability of contact torus bundles

“…which is the proper transform of a complex line and a smooth conic in general position in CP 2 , obtained by blowing up at 4 + n generic points of the conic and one generic point of the complex line, such that the boundary of a closed regular neighborhood of D is −M n . Since the intersection matrix of D is nonsingular and not negative definite, by[LM, Theorem 1.3] (see also…”

Geometry of symplectic log Calabi–Yau pairs

“…Here is a local criterion. The first statement is observed in [12]. Moreover, tori blow-up/down is a local operation that does not change the the diffeomorphism type of Y D and the exactness of ω| Y D .…”

Topological Realizations of Line Arrangements

“…Therefore by [17] the plumbing P A supports a symplectic structure where the core spheres of the plumbing are symplectic, they intersect ω-orthogonally, and there is an inward pointing Liouville vector field demonstrating that the boundary is concave.…”

“…Explicit constructions of symplectic structures on plumbings were first studied by Gay and Stipsicz in [7] to produce symplectic negative definite plumbings with convex boundary. This construction was extended by Li and Mak to the concave case in [17], where it is shown that a plumbing supports a symplectic structure with concave boundary if the plumbing graph satisfies the positive G-S criterion. This means that if Q is the pk`N qˆpk`N q intersection matrix for the plumbing, there exists z P p0, 8q k`N such that Qz is a vector with all positive components.…”