Manifolds which are complex and symplectic but not K\"ahler

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 [$\text{textsc}M\mathrm{.}Fern\acute{a}ndezandV\mathrm{.}Mu\tilde{n}oz$, \emph{An 8-dimensional non-formal simply connected symplectic manifold}, Ann. of Math. (2) $\mathbf{\text{167}}$, no. 3, 1045–1054, 2008.] admits both symplectic and complex structures but does not carry K"ahler metrics.