Strongly Fillable Contact Manifolds and J-holomorphic Foliations

Details Strongly Fillable Contact Manifolds and J-holomorphic Foliations Strongly Fillable Contact Manifolds and J-holomorphic Foliations

2008-06-19

arxiv.org/abs/0806.3193v3

Mathematics

Symplectic Geometry

44 pages, 2 figures; v.3 has a few significant improvements to the main results: We now classify all strong fillings and exact

Scientific paper

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.

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