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 $\omega$. Then we have a natural double complex $\bar{\mathrm{\partial }}+{\bar{\mathrm{\partial }}}^{\mathrm{\Lambda }}$, where ${\bar{\mathrm{\partial }}}^{\mathrm{\Lambda }}$ denotes the symplectic adjoint of the $\bar{\mathrm{\partial }}$-operator. We study the Hard Lefschetz Condition on the Dolbeault cohomology groups of $X$ with respect to the symplectic form $\omega$. In \cite{TW}, we proved that such a condition is equivalent to a certain symplectic analogous of the $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma, namely the $\bar{\mathrm{\partial }}{\bar{\mathrm{\partial }}}^{\mathrm{\Lambda }}$-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 $\overline{\partial}, \overline{\partial}^\Lambda$-Lemmaisstableundersmalldeformationsof$\omega$, but not stable under small deformations of the complex structure. However, if we further assume that $X$ satisfies the $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma then the $\overline{\partial}, \overline{\partial}^\Lambda$-Lemma is stable.