On harmonic symmetries for locally conformally K\"{a}hler manifolds

In this article, we study harmonic symmetries on the compact locally conformally K\"{a}hler manifold $M$ of $dim_{\mathbb{C}}=n$. The space of harmonic symmetries is a subspace of harmonic differential forms which defined by the kernel of a certain Laplacian-type operator $\square$. We observe that the spaces $\ker(\square)\cap\Omega^{l}=\{0\}$ for any $|l-n|\geq2$ and $\ker\Delta_{\bar{\partial}}\cap P^{k,n-1-k}\cap\ker(i_{\theta^{\sharp}})\cong\ker(\square^{k,n-1-k})$, $\ker\Delta_{\bar{\partial}}\cap P^{k,n-k}\cong\ker(\square^{k,n-k})$. Furthermore, suppose that $M$ is a Vaisman manifold, we prove that (i) $\alpha$ is $(n-1)$-form in $\ker(\square)$ if only if $\alpha$ is a transversally harmonic and transversally effective $\mathcal{V}$-foliate form; (ii) $\alpha$ is a $(p,n-p)$-form in $\ker(\square^{p,n-p})$ if only if there are two forms $\beta_{1}\in\mathcal{S}^{p-1,n-p}$ and $\beta_{2}\in\mathcal{S}^{p,n-p-1}$ such that $\alpha=\theta^{1,0}\wedge\beta_{1}+\theta^{0,1}\wedge\beta_{2}$.