On harmonic symmetries for locally conformally Kähler manifolds

Abstract

In this article, we study harmonic symmetries on the compact locally conformally Kähler 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 $\square$. We observe that the spaces $\ker (\square )\cap {\Omega }^l=\{0\}$ for any $|l-n|\ge 2$ and $\ker {\Delta }_{\bar{\partial }}\cap P^{k,n-1-k}\cap \ker (i_{{\theta }^{♯}})\cong \ker ({\square }^{k,n-1-k})$, $\ker {\Delta }_{\bar{\partial }}\cap P^{k,n-k}\cong \ker ({\square }^{k,n-k})$. 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 $\alpha ={\theta }^{1,0}\wedge {\beta }_1+{\theta }^{0,1}\wedge {\beta }_2$.

19 pages, to appear in Annali di Matematica Pura ed Applicata (1923-)