Unobstructed symplectic packing by ellipsoids for tori and hyperkahler manifolds

TL;DR

This paper proves that symplectic packings by ellipsoids are always possible in certain complex and hyperkähler manifolds, extending known results to a broader class of manifolds.

Contribution

It establishes unobstructed symplectic packings by ellipsoids for all even-dimensional tori with Kahler forms and hyperkähler manifolds of maximal holonomy, including limits of Campana simple manifolds.

Findings

Symplectic packings by ellipsoids are unobstructed for all even-dimensional tori with Kahler forms.

Unobstructed packings are also valid for all closed hyperkähler manifolds of maximal holonomy.

The proof uses Kahler resolutions and results on Kahler cohomology classes.

Abstract

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as well as for any closed Kahler manifold which is a limit of Campana simple manifolds in a smooth deformation. The proof involves the construction of a Kahler resolution of a Kahler orbifold with isolated singularities and relies on the results of Demailly-Paun and Miyaoka on Kahler cohomology classes.