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Working paper

Full symplectic packing for tori and hyperkahler manifolds

Entov M., Verbitsky M.
Let M be a closed symplectic manifold of volume V. We say that M admits a full symplectic packing by balls if any collection of symplectic balls 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 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, admits a full 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 full 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).