Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

TL;DR

This paper constructs a simply-connected compact complex non-Kähler manifold with a balanced metric satisfying the -Lemma, using -cohomology to study blow-ups and their properties.

Contribution

It introduces a new approach employing -cohomology to analyze the stability of the -Lemma under blow-ups of complex manifolds.

Findings

Constructed a non-Khler manifold satisfying the -Lemma.

Provided a -cohomology framework for blow-up analysis.

Demonstrated stability of -Lemma under certain modifications.

Abstract

We construct a simply-connected compact complex non-K\"ahler 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 \v{C}ech 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.