Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

We construct a simply-connected compact complex non-Kahler manifold satisfying the partial derivative(partial derivative) over bar -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 derivative(partial derivative) over bar -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 tech cohomology theory to study the Dolbeault cohomology of the blow-up (X) over tilde (Z) of a compact complex manifold X along a submanifold Z admitting a holomorphically contractible neighbourhood.