A geometric criterion for prescribing residues and some applications

Abstract

Let X be a compact complex manifold and D a ℂ-linear finite formal sum of divisors of X. A theorem of Weil and Kodaira says that if X is Kähler, then 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,ℂ). We generalized their theorem to general compact complex manifolds. The necessary and sufficient condition is described by a new invariant called 𝒬-flat class. In the second part of the paper, we classify all the pluriharmonic functions on a compact algebraic manifold with mild singularities.