Symplectic almost squeezings

Abstract:  Around 2000, Biran introduced the notion of polarization of a symplectic manifold, and showed that the associated Lagrangian skeleta exhibit remarkable rigidity properties. He proved in particular that their complements may have small Gromov width. In this work, we introduce a version of polarization on affine symplectic manifolds. These polarizations are more flexible than those of closed symplectic manifolds, which provides a wider range of applications. For instance, given an affine symplectic manifold V and any closed symplectic 4-manifold M of larger volume, there exists an isotropic CW complex in V such that its complement symplectically embeds into M. Specifically, after removing from a 4-ball of any radius finitely Lagrangian planes, one finds an embedding into the standard cylinder,  extending a result by Sackel-Song-Varolgunes-Zhu and Brendel. 

This is work joint with Emmanuel Opshtein.