论文标题
关于本地保态的谐波对称性Kähler歧管
On harmonic symmetries for locally conformally Kähler manifolds
论文作者
论文摘要
在本文中，我们研究了紧凑型在本地的谐波对称性，kähler歧管$ m $ of $ dim _ {\ mathbb {c}}} = n $。谐波对称的空间是谐波差分形式的子空间，该子空间由特定拉普拉斯型操作员$ \ square $的内核定义。我们观察到，对于任何$ | l-n | \ geq2 $和$ \kerδ_ {\ bar {\ partial}} \ cap p^{k，n-1-k} \ cap \ ker（i_ {θ^{\ sharp}}）\ cong \ ker（\ square^{k，n-1-k}）$，$ \kerΔ__{\ bar {\ bar {\ partial}}} \ cap p^{k，n-k} \ cong \ ker（\ square^{k，n-k}）$。此外，假设$ m $是VAISMAN的歧管，我们证明（i）$α$是$（n-1）$ - $ \ ker（\ square）$的形式，仅当$α$是横向谐波和横向有效$ \ $ \ $ \ Mathcal {v} $ - foliate-foliate-foliate-foliate-foliate-foliate-foliate-foliate-foliate-fore; （ii）$α$是$ \ ker（\ square^{p，n-p}）$中的$（p，n-p）$ - 仅当有两种形式$β_{1} \ in \ in \ in \ mathcal {s}^{s}^{p-1，n-p}，n-p} $和$β_{2} $ n \ n \ n \ n \ n \ n \ n s} s} $α=θ^{1,0} \wedgeβ_{1}+θ^{0,1} \wedgeβ_{2} $。
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 $α=θ^{1,0}\wedgeβ_{1}+θ^{0,1}\wedgeβ_{2}$.
