Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

• Main Authors: Angella Daniele, Suwa Tatsuo, Tardini Nicoletta, Tomassini Adriano
• Format: Article
• Language: English
• Published: De Gruyter 2020-09-01
• Series: Complex Manifolds
• Subjects: complex manifold; non-kähler geometry; ∂ ̅∂-lemma; hodge decomposition; modification; blowup; dolbeault cohomology; orbifold; 32q99; 32c35; 32s45
• Online Access: https://doi.org/10.1515/coma-2020-0103
• ISSN: 2300-7443

We construct a simply-connected compact complex non-Kähler manifold satisfying the ∂ ̅∂ -Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the ∂ ̅∂-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in [34, 39, 40, 50] with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blowup ̃XZ of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.