Kähler-type embeddings of balls into symplectic manifolds (Q6601489)

In this paper, in an attempt to get more information about the connectedness question for symplectic manifolds of dimension higher than 4, the authors consider a more restrictive class of symplectic embeddings of disjoint unions of domains. They extend the results of \textit{D. McDuff} and \textit{L. Polterovich} [Invent. Math. 115, No. 3, 405--429 (1994; Zbl 0833.53028)] to Kähler-type embeddings of closed balls \(\sqcup_{i=1}^kB(r_i)\) of radii \(r_1,\dots,r_k\). \N\NConsider a symplectic embedding of a disjoint union of domains (lying in the standard symplectic \(\mathbb{R}^{2n}\)) into a symplectic manifold \(M\). Such an embedding is of Kähler-type, respectively tame, if it is holomorphic with respect to some (not a priori fixed) Kähler-type integrable complex structure on \(M\) compatible with the symplectic form, respectively tamed by it. Assume that \(M\) 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 Kähler-type symplectic form. Then any two Kähler-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 Kähler-type complex structures on \(M\), then the symplectomorphism can be chosen to be smoothly isotopic to the identity. \N\NFor certain \(M\) and certain disjoint unions of balls, the authors describe precisely the obstructions to the existence of Kähler-type embeddings of the balls into \(M\). In particular, symplectic volume is the only obstruction for the existence of Kähler-type embeddings of ln equal balls (for any l) into \(\mathbb{C}P^n\) 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. They 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 Kähler-type symplectic form. For balls and parallelepipeds the same is true in \(K3\) surfaces. \N\NThe paper consists of the twelve chapters and is supported by an appendix about dependence of the Hodge decomposition on the complex structure, deformation families of complex structures, relative version of Moser's method and Alexander's trick.