Unobstructed symplectic packing by ellipsoids for tori and hyperkähler manifolds

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 Kähler symplectic forms and all closed hyperkähler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kähler manifolds that are not unions of their complex subvarieties), as well as for any closed Kähler manifold which is a limit of Campana simple manifolds in a smooth deformation. The proof involves the construction of a Kähler resolution of a Kähler orbifold with isolated singularities and relies on the results of Demailly–Paun and Miyaoka on Kähler cohomology classes.