$L^{2}$-Hodge theory on complete almost Kähler manifold and its application (2302.14032v3)

Abstract: Let $(X,J,\omega)$ be a complete $2n$-dimensional almost K\"{a}hler manifold. First part of this article, we construct some identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of $\ker{\Delta_{\partial}}\cap\ker{\Delta_{\bar{\partial}}}$ on pure bidegree. In the second part, as some applications of those identities, we establish some vanishing theorems on the spaces of $L^{2}$-harmonic $(p,q)$-forms on $X$ under some growth assumptions on the K\"{a}her form $\omega$. We also give some $L^{2}$-estimates to sharpen the vanishing theorems in two specific cases. At last of the article, as an application, we study the topology of the compact almost K\"{a}hler manifold with negative sectional curvature.