Manifolds which are complex and symplectic but not Kähler

The first example of a compact manifold admitting both complex and symplectic structures but not admitting a K"ahler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply-connected 8-dimensional compact manifold of [ extscM. Fern'andez and V. Mu~noz, emphAn 8-dimensional non-formal simply connected symplectic manifold, Ann. of Math. (2) extbf167, no. 3, 1045--1054, 2008.] admits both symplectic and complex structures but does not carry K"ahler metrics.