Abstract
AbstractWe 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.
Highlights
The ∂∂-Lemma is a strong cohomological decomposition property de ned for complex manifolds, which is satis ed for example by algebraic projective manifolds and, more generally, by compact Kähler manifolds
We were initially aimed at investigating the stability of the property of satisfying the ∂∂-Lemma under modi cations of compact complex manifolds and orbifolds
This property yields strong topological obstructions: the real homotopy type of a compact complex manifold satisfying the ∂∂-Lemma is a formal consequence of its cohomology ring [16]
Summary
The ∂∂-Lemma is a strong cohomological decomposition property de ned for complex manifolds, which is satis ed for example by algebraic projective manifolds and, more generally, by compact Kähler manifolds. By the results contained in [13, Corollary 3.13], [28, Theorem 1] and thanks to the stability property of the ∂∂-Lemma for small deformations [47, Proposition 9.21], [52, Theorem 5.12] one can produce examples of compact complex manifolds satisfying the ∂∂-Lemma and not bimeromorphic to Kähler manifolds. Other examples of this kind can be found among solvmanifolds [4, 5, 26]; other examples are provided by Clemens manifolds [18, 19], which are constructed by combining modi cations and deformations
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