A $6$-dimensional simply connected complex and symplectic manifold with no Kähler metric

Abstract

We construct a simply connected compact manifold which has complex and symplectic structures but does not admit Kähler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form.

14 pages, no figures. Second version: minor changes in the introduction, exposition clarified and new references added. Comments are welcome!