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Cohomologies of complex manifolds with symplectic $(1,1)$-forms

Let $(X, J)$ be a complex manifold with a non-degenerated smooth $d$-closed $(1,1)$-form $ω$. Then we have a natural double complex $\overline{\partial}+\overline{\partial}^Λ$, where $\overline{\partial}^Λ$ denotes the symplectic adjoint of the $\overline{\partial}$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $ω$. In \cite{TW}, we proved that such a condition is equivalent to a certain symplectic analogous of the $\partial\overline{\partial}$-Lemma, namely the $\overline{\partial}\, \overline{\partial}^Λ$-Lemma, which can be characterized in terms of Bott--Chern and Aeppli cohomologies associated to the above double complex. We obtain Nomizu type theorems for the Bott--Chern and Aeppli cohomologies and we show that the $\overline{\partial}\, \overline{\partial}^Λ$-Lemma is stable under small deformations of $ω$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $\partial\overline{\partial}$-Lemma then the $\overline{\partial}\, \overline{\partial}^Λ$-Lemma is stable.