Kahler-type embeddings of balls into symplectic manifolds

A symplectic embedding of a disjoint union of balls into a symplectic manifold M is said to be of Kahler type if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M compatible with the symplectic form. I'll discuss when such embeddings into a closed symplectic manifold exist and when two of them can be mapped into each other by a symplectomorphism.

This is a joint work with M.Verbitsky.