On the ${\mathcal{D}}_J^{+}$ operator on higher-dimensional almost Kähler manifolds

In this paper, we introduce ${\mathcal{D}}_J^{+}$, a generalization of $\mathrm{\partial }\bar{\mathrm{\partial }}$ operator on higher dimensional almost Kähler manifolds. Using the ${\mathcal{D}}_J^{+}$ operator, we investigate the $\bar{\mathrm{\partial }}$-problem in almost Kähler geometry and explore the generalized Monge-Ampère equation on almost Kähler manifolds. We establish a uniqueness up to the addition of a constant and local existence theorem for this equation. At last, we find an elliptical system for ${\mathcal{D}}_J^{+}$ operator. As an application, we reorganize the result of Tosatti-Weinkove-Yau in \cite{TWY}.