abstract: The goal of this talk is to give an overview of the existing examples of manifolds which are symplectic and complex but carry no Kähler metric. I will also construct an example of a simply connected manifold which admits both symplectic and complex structures, but no Kähler structure. Such a manifold has dimension 6, the lowest in which such a phenomenon can occurr.
References
{BFM} G. Bazzoni, M. Fernández and V. Muñoz, A 6-dimensional simply
connected complex and symplectic manifold with no Kähler metric,
http://arxiv.org/abs/1410.6045, preprint.