Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

We construct a simply-connected compact complex non-K"ahler manifold satisfying the $partialoverlinepartial$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $partialoverlinepartial$-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 {C}ech cohomology theory to study the Dolbeault cohomology of the blow-up $ ilde X_Z$ of a compact complex manifold $X$ along a submanifold $Z$ admitting a holomorphically contractible neighbourhood.