On Weinstein domains in symplectic manifolds

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.