Weak symplectic fillings and holomorphic curves

Niederkrüger, Klaus; Wendl, Chris

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 5, pp. 801-853.

On montre plusieurs résultats concernant les remplissages faibles de variétés de contact de dimension $3$, notamment : (1) Les remplissages faibles des variétés de contact planaires sont à déformation près des éclatements de remplissages de Stein. (2) Les variétés de contact ayant de la torsion planaire et satisfaisant une certaine condition homologique n’admettent pas de remplissages faibles - de cette manière on obtient des nouveaux exemples de variétés de contact qui ne sont pas faiblement remplissables. (3) La remplissabilité faible est préservée par l’opération de somme connexe le long de tores pré-lagrangiens - ce qui nous donne beaucoup de nouveaux exemples de variétés de contact sans torsion de Giroux qui sont faiblement, mais pas fortement, remplissables. On établit une obstruction à la remplissabilité faible avec deux approches qui utilisent des courbes holomorphes. La première méthode se base sur l’argument original de Gromov-Eliashberg des « disques de Bishop » . On utilise une famille d’anneaux holomorphes s’appuyant sur un « anneau vrillé ancré » pour étudier le cas spécial de la torsion de Giroux. La deuxième méthode utilise des courbes holomorphes à pointes, et elle se base sur l’observation que, dans un remplissage faible, la structure symplectique peut être déformée au voisinage du bord, en une structure hamiltonienne stable. Cette observation permet aussi d’appliquer les méthodes à la théorie symplectique de champs, et on montre dans un cas simple que la distinction entre les remplissabilités faible et forte se traduit en homologie de contact par une distinction entre coefficients tordus et non tordus.

We prove several results on weak symplectic fillings of contact $3$-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable-this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori-this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fillable. We establish the obstructions to weak fillings via two parallel approaches using holomorphic curves. In the first approach, we generalize the original Gromov-Eliashberg “Bishop disk” argument to study the special case of Giroux torsion via a Bishop family of holomorphic annuli with boundary on an “anchored overtwisted annulus”. The second approach uses punctured holomorphic curves, and is based on the observation that every weak filling can be deformed in a collar neighborhood so as to induce a stable Hamiltonian structure on the boundary. This also makes it possible to apply the techniques of Symplectic Field Theory, which we demonstrate in a test case by showing that the distinction between weakly and strongly fillable translates into contact homology as the distinction between twisted and untwisted coefficients.

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