A geometric criterion for prescribing residues and some applications

Abstract

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.