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

2023/03/07 by Kasuya, Hisashi, Stelzig, Jonas
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2303.03641

Abstract

We give four constructions of non-∂∂ (hence non-Kähler) manifolds: (1) A simply connected page-1-∂∂-manifold (2) A simply connected dd^c+3-manifold (3) For any r≥ 2, a simply connected compact manifold with nonzero differential on the r-th page of the Frölicher spectral sequence. (4) For any r≥ 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.

Related

• Cohomological aspects of non-Kähler manifolds
• Higher-Page Hodge Theory of Compact Complex Manifolds
• Manifolds which are complex and symplectic but not Kähler
• Coeffective cohomology of symplectic aspherical manifolds
• ∂∂-Lemma and p-Kähler structures on families of solvmanifolds