Kahler-type embeddings of balls into symplectic manifolds

Abstract: A symplectic embedding of a disjoint union of balls into a symplectic manifold M is called Kahler-type if it is holomorphic with respect to some (not a priori fixed) complex structure on M compatible with the symplectic form. Assume that M either of the following: CP^n (with the standard symplectic form), an even-dimensional torus or a K3 surface equipped with an irrational Kahler-type symplectic form. Then:

1. 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 the 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.

2. Symplectic volume is the only obstruction for the existence of Kahler-type embeddings of k^n equal balls (for any k) into CP^n and of any number of possibly different balls into a torus or a K3 surface. This is a joint work with M.Verbitsky.

differential geometrydynamical systemsgeometric topologysymplectic geometryspectral theory

Audience: researchers in the topic

Geometry and Dynamics seminar

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