一个确定残数的几何判据及其应用
A geometric criterion for prescribing residues and some applications
摘要：
Weil和Kodaira的一个老定理说，对于紧致Kähler流形$X$，存在一个闭对数$1$ -形式，其剩余因子为$D$当且仅当$D$在$H_{2n-2}(X,\mathbb C)$中同调于零。在本文的第一部分中，我们推广上述定理一般紧复流形的充要条件一般是由称为$\mathcal Q$ -平坦类的全纯不变量描述。其次，证明了当$X$具有性质$(H)$时，全纯判据可化为拓扑判据。由于所有凯勒流形都有性质$(H)$，这给出了韦尔和小平原始定理的另一种证明。然后，我们利用上述一般定理证明了闭亚纯$1$ -形式的一些分解定理。第二部分研究了射影流形上的多重调和函数，并对所有具有轻度奇异性的多重调和函数进行了分类。
英文摘要：
An old theorem of Weil and Kodaira says that for a compact Kähler manifold $X$ there is a closed logarithmic $1$-form with residue divisor $D$ if and only if $D$ is homologous to zero in $H_{2n-2}(X,\mathbb C)$. In the first part of this paper, we generalize the above theorem to general compact complex manifolds by showing that the necessary and sufficient condition in general is described by a holomorphic invariant called the $\mathcal Q$-flat class. Next, we prove that the holomorphic criterion is reduced to the topological one when $X$ has Property $(H)$. Since all Kähler manifolds have Property $(H)$, this gives an alternative proof of Weil and Kodaira's original theorem. Then, we prove some decomposition theorems for closed meromorphic $1$-forms by applying the above general theorem. In the second part of the paper, we turn to the study of pluriharmonic functions on projective manifolds and classify all the pluriharmonic functions with mild singularity.
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