On $L^{2}$-harmonic forms of complete almost K\"{a}hler manifold

In this article, we study the $L^{2}$-harmonic forms on the complete $2n$-dimensional almost K\"{a}her manifold $X$. We observe that the $L^{2}$-harmonic forms can decomposition into Lefschetz powers of primitive forms. Therefore we can extend vanishing theorems of $d$(bounded) (resp. $d$(sublinear)) K\"{a}hler manifold proved by Gromov (resp. Cao-Xavier, Jost-Zuo) to almost K\"{a}hlerian case, that is, the spaces of all harmonic $(p,q)$-forms on $X$ vanishing unless $p+q=n$. We also give a lower bound on the spectra of the Laplace operator to sharpen the Lefschetz vanishing theorem on $d$(bounded) case.