Unobstructed symplectic packing for tori and hyperkahler manifolds

TL;DR

This paper proves that certain classes of Kahler and hyperkahler manifolds, including tori and limits of Campana simple manifolds, admit unobstructed symplectic packings by balls, extending previous results and using advanced geometric theorems.

Contribution

It establishes unobstructed symplectic packings for Campana simple Kahler manifolds and their limits, including tori and hyperkahler manifolds, generalizing prior work.

Findings

Campana simple Kahler manifolds admit unobstructed packings

All even-dimensional tori with Kahler forms admit packings by balls

Tori with non-rational Kahler classes admit packings by polydisks

Abstract

Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun structure theorem. We prove that any Campana simple Kahler manifold, as well as any manifold which is a limit of Campana simple manifolds in a smooth deformation, admits an unobstructed symplectic packing by balls. This is used to show that all even-dimensional tori equipped with Kahler symplectic forms and all hyperkahler manifolds of maximal holonomy admit unobstructed symplectic packings by balls. This generalizes a previous result by Latschev-McDuff-Schlenk. We also consider symplectic packings by other shapes and show, using Ratner's orbit closure theorem, that any even-dimensional torus equipped with a Kahler form whose cohomology class is not proportional to a rational one admits a full symplectic packing by any number of equal polydisks (and, in particular, by any number of equal cubes).