Kahler-type Embeddings of Balls into Symplectic Manifolds

Consider a symplectic embedding of a disjoint union of domains (lying in the standard symplectic R²ⁿ) into a symplectic manifold M. We say that such an embedding is Kahlertype, or respectively tame, if it is holomorphic with respect to some (not a priori fixed, Kahler-type) integrable complex structure on M compatible with the symplectic form, or respectively tamed by it. Assume thatM is either of the following: A complex projective space (with the standard symplectic form); an even-dimensional torus, or a K3 surface, equipped with an irrational Kahler-type symplectic form. Then any two Kahler-type embeddings of a disjoint union of balls into M can be mapped into each other by a symplectomorphism acting trivially on homology. If the embeddings are holomorphic with respect to complex structures compatible with the symplectic form and lying in the same connected component of the space of Kahler-type complex structures on M, then the symplectomorphism can be chosen to be smoothly isotopic to the identity. For certain M and certain disjoint unions of balls we describe precisely the obstructions to the existence of Kahler-type embeddings of the balls into M. In particular, symplectic volume is the only obstruction for the existence of Kahler-type embeddings of lⁿ equal balls (for any l) into CPⁿ with the standard symplectic form and of any number of possibly different balls into a torus or a K3 surface, equipped with an irrational symplectic form. We also show that symplectic volume is the only obstruction for the existence of tame embeddings of disjoint unions of equal balls, polydisks, or parallelepipeds, into a torus equipped with a generic Kahler-type symplectic form. For balls and parallelepipeds the same is true for K3 surfaces.