  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 connected
compact manifold with nonzero differential on the $r$-th page of the
Fr\"olicher spectral sequence. (4) For any $r\geq 2$, a pluriclosed nilmanifold
with nonzero differential on the $r$-th page of the Fr\"olicher 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.
