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 $di{m}_{\mathbb{C}}=n$. The space of harmonic symmetries is a subspace of harmonic differential forms which defined by the kernel of a certain Laplacian-type operator $\mathrm{\square }$. We observe that the spaces $\ker (\mathrm{\square })\cap {\mathrm{\Omega }}^l=\{0\}$ for any $\mathrm{\mid }l-n\mathrm{\mid }\ge 2$ 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 $\ker (\mathrm{\square })$ if only if $\alpha$ is a transversally harmonic and transversally effective $\mathcal{V}$-foliate form; (ii) $\alpha$ is a $(p,n-p)$-form in $\ker ({\mathrm{\square }}^{p,n-p})$ 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 $\alpha ={\theta }^{1,0}\wedge {\beta }_1+{\theta }^{0,1}\wedge {\beta }_2$.