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arXiv · 2503.14101

On the $\mathcal{D}^+_J$ operator on higher-dimensional almost K\"{a}hler manifolds

Qiang Tan·Hongyu Wang·Ken Wang·Zuyi Zhang
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Abstract

In this paper, we introduce $\mathcal{D}^+_J$, a generalization of $\partial\bar{\partial}$ operator on higher dimensional almost K\"{a}hler manifolds. Using the $\mathcal{D}^+_J$ operator, we investigate the $\bar{\partial}$-problem in almost K\"{a}hler geometry and explore the generalized Monge-Amp\`{e}re equation on almost K\"{a}hler 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}.

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Qiang Tan, Hongyu Wang, Ken Wang, Zuyi Zhang. 2025-03-18. On the $\mathcal{D}^+_J$ operator on higher-dimensional almost K\"{a}hler manifolds. https://arxiv.org/abs/2503.14101

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