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

TL;DR

This paper investigates harmonic symmetries on compact locally conformally Kähler manifolds, characterizing the kernel of a Laplacian-type operator and exploring special cases like Vaisman manifolds, revealing new geometric and analytical properties.

Contribution

It introduces a detailed analysis of harmonic symmetries on locally conformally Kähler manifolds, including characterizations on Vaisman manifolds, and establishes new relationships between harmonic forms and geometric structures.

Findings

Kernel of the Laplacian-type operator vanishes outside specific degrees.

Characterization of harmonic forms on Vaisman manifolds in terms of transversally harmonic forms.

Explicit description of harmonic forms in terms of wedge products with the Lee form.

Abstract

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 $\alpha=\theta^{1,0}\wedge\beta_{1}+\theta^{0,1}\wedge\beta_{2}$.