Let (X, J) be a 4-dimensional compact almost-complex manifold and let g be a Hermitian metric on (X, J). Denote by ∆If g is globally conformally Kähler, respectively (strictly) locally conformally Kähler, we prove that the dimension of the space of ∂-harmonic (1, 1)-forms on X is a topological invariant given by b − + 1, respectively b − . As an application, we provide a one-parameter family of almost-Hermitian structures on the Kodaira-Thurston manifold for which such a dimension is b − . This gives a positive answer to a question raised by T. Holt and W.Zhang.

We consider several differential operators on compact almostcomplex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.2010 Mathematics Subject Classification. 32Q60, 53C15, 58A14, 53D05. . She is grateful to the department for the hospitality. PreliminariesLet (X , J) be an almost-complex manifold then the almost-complex structure J induces a natural bi-graduation on the space of forms A • (X) = p+q=• A p,q (X). If J is non-integrable the exterior derivative d acts on forms as

Dedicated to the memory of Professor Pierre Dolbeault.Abstract. We introduce a "qualitative property" for Bott-Chern cohomology of complex non-Kähler manifolds, which is motivated in view of the study of the algebraic structure of Bott-Chern cohomology. We prove that such a property characterizes the validity of the ∂∂-Lemma. This follows from a quantitative study of Bott-Chern cohomology. In this context, we also prove a new bound on the dimension of the Bott-Chern cohomology in terms of the Hodge numbers. We also give a generalization of this upper bound, with applications to symplectic cohomologies.

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.