On Weinstein domains in symplectic manifolds

TL;DR

This paper proves that Weinstein domains embedded in closed symplectic manifolds always have symplectic hypersurfaces in their complement, providing new obstructions for certain 3-manifolds to bound Weinstein domains in specific 4-manifolds.

Contribution

It introduces a method to find symplectic hypersurfaces in the complement of Weinstein domains and establishes obstructions for certain 3-manifolds to bound Weinstein domains in rational surfaces.

Findings

No Brieskorn homology sphere bounds a Weinstein domain in certain rational surfaces.

Existence of Weinstein domains in some positive symplectic rational surfaces for k ≥ 8.

Several Brieskorn spheres do not bound Weinstein domains in any positive symplectic 4-manifold.

Abstract

We prove that a Weinstein domain symplectically embedded in a closed symplectic manifold always admits symplectic hypersurfaces in its complement, possibly after a deformation. As a consequence, we obtain an obstruction for a closed 3-dimensional manifold to arise as the boundary of a Weinstein domain in a class of symplectic 4-manifolds that includes many symplectic rational surfaces. A particular application is that no Brieskorn homology sphere bounds a Weinstein domain symplectically embedded in a rational surface diffeomorphic to $S^2\times S^2$ or to ${\mathbb C} P^2\# k \overline{{\mathbb C}P}^2$, for any $k\leq 7$, despite the fact that many Brieskorn spheres bound Stein domains holomorphically embedded in these rational surfaces. Several families of Brieskorn spheres are obtained that do not bound a Weinstein domain in any 4-manifold with a ``positive'' symplectic structure. Such Weinstein domains do exist in certain positive symplectic rational surfaces when $k\geq 8$, though their topology is significantly constrained.