Symplectic divisors in dimension four

Abstract

We study the symplectic and contact geometry related to symplectic divisors in symplectic 4-manifolds. We start by showing the contact structure induced on the boundary of a divisor neighborhood is invariant under toric and interior blow-ups and blow-downs. We also construct an open book decomposition on the boundary of a concave divisor neighborhood and apply it to the study of universally tight contact structures of contact torus bundles. Next, we classify, up to toric equivalence, all concave circular spherical divisors D that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such D. When D is anticanonical and convex, we give explicit betti number bounds for Stein fillings of its boundary contact torus bundle. Finally we study the moduli space of symplectic log Calabi-Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and study its relation with various other moduli spaces. In particular, we introduce the notion of toric symplectic log Calabi-Yau divisors and relate it to toric actions. Then we derive an upper bound for the count of symplectic log Calabi-Yau divisors and give an exact count in the case of 2- and 3-point blow-ups of complex projective space. Along the way, we also prove a stability result for symplectic log Calabi-Yau divisors, which might be of independent interest.