Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

Abstract

We construct a simply-connected compact complex non-Kähler manifold satisfying the $\partial \bar{\partial }$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $\partial \bar{\partial }$-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in \cite{rao-yang-yang, yang-yang, stelzig-blowup, stelzig-doublecomplex} with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blow-up ${\tilde{X}}_Z$ of a compact complex manifold $X$ along a submanifold $Z$ admitting a holomorphically contractible neighbourhood.

References in corpus (7)

• Autour de la cohomologie de Bott-Chern
• The Double Complex of a Blow-up
• Volume and Self-Intersection of Differences of Two Nef Classes
• Sato hyperfunctions via relative Dolbeault cohomology
• The $\partial \bar{\partial }$-lemma for general Clemens manifolds
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Cited by in corpus (9)

• On the Structure of Double Complexes
• Dolbeault and Bott-Chern formalities: deformations and $\partial \overline{\partial }$-lemma
• Hypercohomologies of truncated twisted holomorphic de Rham complexes
• Dolbeault cohomologies of blowing up complex manifolds
• On the deformed Bott-Chern cohomology
• Relative Dolbeault cohomology
• The $\partial \bar{\partial }$-lemma under surjective maps
• The heredity and bimeromorphic invariance of the $\partial \overline{\partial }$-lemma property
• Blow-up formulae for twisted cohomologies with supports