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

Abstract

We give four constructions of non-$\partial \bar{\partial }$ (hence non-Kähler) manifolds: (1) A simply connected page-$1$-$\partial \bar{\partial }$-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.

The proof of Theorem A in v1 had a gap pointed out to us by D. Abramovich. In v2 we use a more elementary resolution procedure to prove Theorem A and the statement is less general (but still sufficient for all constructions given). New title and further minor changes. 21 pages, 2 figures, 2 tables. Comments welcome!