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arXiv · 2303.03641

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

Hisashi Kasuya·Jonas Stelzig
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Abstract

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.

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Hisashi Kasuya, Jonas Stelzig. 2023-03-07. Complex non-K\"ahler manifolds that are cohomologically close to, or far from, being K\"ahler. https://arxiv.org/abs/2303.03641

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