Dolbeault and J-Invariant Cohomologies on Almost Complex Manifolds

In this paper we relate the cohomology of J-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.

Dolbeault cohomology of complex manifolds with torus action

We describe the basic Dolbeault cohomology algebra of the canonical foliation on a class of complex manifolds with a torus symmetry group. This class includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most generality, complex manifolds with a maximal holomorphic torus action. We also provide a DGA model for the ordinary Dolbeault cohomology algebra. The Hodge decomposition for the basic Dolbeault cohomology is proved by reducing to the transversely Kähler (equivalently, polytopal) case using a foliated analogue of toric blow-up.

Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

We 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.