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We give four constructions of non-$\partial\bar\partial$ (hence non-K\"ahler)manifolds: (1) A simply connected page-$1$-$\partial\bar\partial$-manifold (2)A simply connected $dd^c+3$-manifold (3) For any $r\geq 2$, a simply connectedcompact manifold with nonzero differential on the $r$-th page of theFr\"olicher spectral sequence. (4) For any $r\geq 2$, a pluriclosed nilmanifoldwith nonzero differential on the $r$-th page of the Fr\"olicher spectralsequence. The latter disproves a conjecture by Popovici. A main ingredient inthe first three constructions is a simple resolution construction of certainquotient singularities with control on the cohomology.

Complex non-K\"ahler manifolds that are cohomologically close to, or far from, being K\"ahler