Complex non-Kähler manifolds that are cohomologically close to, or far from, being Kähler

We give four constructions of non-$\mathrm{\partial }\bar{\mathrm{\partial }}$ (hence non-Kähler) manifolds: (1) A simply connected page-$1$-$\mathrm{\partial }\bar{\mathrm{\partial }}$-manifold (2) A simply connected $d{d}^c+3$-manifold (3) For any $r\ge 2$, a simply connected compact manifold with nonzero differential on the $r$-th page of the Frölicher spectral sequence. (4) For any $r\ge 2$, a pluriclosed nilmanifold with nonzero differential on the $r$-th page of the Frölicher spectral sequence. The latter disproves a conjecture by Popovici. A main ingredient in the first three constructions is a simple resolution construction of certain quotient singularities with control on the cohomology.