On harmonic symmetries for locally conformally Kähler manifolds

In this article, we study harmonic symmetries on the compact locally conformally Kähler 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 $\ker(\square)\capΩ^{l}=\{0\}$ for any $|l-n|\geq2$ and $\kerΔ_{\bar{\partial}}\cap P^{k,n-1-k}\cap\ker(i_{θ^{\sharp}})\cong\ker(\square^{k,n-1-k})$, $\kerΔ_{\bar{\partial}}\cap P^{k,n-k}\cong\ker(\square^{k,n-k})$. Furthermore, suppose that $M$ is a Vaisman manifold, we prove that (i) $α$ is $(n-1)$-form in $\ker(\square)$ if only if $α$ is a transversally harmonic and transversally effective $\mathcal{V}$-foliate form; (ii) $α$ is a $(p,n-p)$-form in $\ker(\square^{p,n-p})$ if only if there are two forms $β_{1}\in\mathcal{S}^{p-1,n-p}$ and $β_{2}\in\mathcal{S}^{p,n-p-1}$ such that $α=θ^{1,0}\wedgeβ_{1}+θ^{0,1}\wedgeβ_{2}$.

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