Strongly Fillable Contact Manifolds and J-holomorphic Foliations

TL;DR

This paper demonstrates that strong symplectic fillings of certain contact manifolds admit Lefschetz fibrations, establishing new connections between fillability, Stein structures, and J-holomorphic curve foliations, with implications for symplectomorphism groups and fillability obstructions.

Contribution

It proves the existence of Lefschetz fibrations for strong fillings of planar contact manifolds and the 3-torus, linking fillability to Stein structures and symplectic deformation classes.

Findings

Strong fillings of planar contact manifolds admit Lefschetz fibrations.

Strong fillings of the 3-torus are equivalent up to deformation and blowup.

A new obstruction to strong fillability is introduced, providing a non-gauge-theoretic proof of nonfillability results.

Abstract

We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of the 3-torus similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplectomorphism group on the cotangent bundle of the 2-torus is contractible, and to define an obstruction to strong fillability that yields a non-gauge-theoretic proof of Gay's recent nonfillability result for contact manifolds with positive Giroux torsion.