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

TL;DR

This paper investigates the structure and vanishing properties of $L^{2}$-harmonic forms on complete almost K"{a}hler manifolds, extending classical results from K"{a}hler geometry to a broader setting.

Contribution

It extends vanishing theorems for harmonic forms from K"{a}hler to almost K"{a}hler manifolds and provides spectral bounds for the Laplace operator.

Findings

$L^{2}$-harmonic forms decompose into Lefschetz powers of primitive forms.

Vanishing of harmonic $(p,q)$-forms unless $p+q=n$ on certain almost K"{a}hler manifolds.

Lower bounds on Laplace spectrum to refine Lefschetz vanishing results.

Abstract

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.