Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

Abstract

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.

Keywords

• complex manifold
• non-kähler geometry
• ∂ ̅∂-lemma
• hodge decomposition
• modification
• blowup
• dolbeault cohomology
• orbifold
• 32q99
• 32c35
• 32s45