A geometric criterion for prescribing residues and some applications

Fang, Hanlong

Annales de l'Institut Fourier, Tome 71 (2021) no. 5, pp. 1963-2018

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.

Soit $X$ une variété complexe compacte et $D$ une somme formelle finie $ℂ$-linéaire des diviseurs de $X$. Un théorème de Weil et Kodaira dit que si $X$ est kählerienne, alors il existe une $1-$forme logarithmique fermé avec un diviseur résiduel $D$ si et seulement si $D$ est homologue à zéro dans $H_{2n-2}(X,ℂ)$. Nous généralisons leur théorème aux variétès complexes compactes générales. La condition nécessaire et suffisante est décrite par un nouvel invariant appelé $𝒬$-flat class. Dans la deuxième partie de l’article, nous classons toutes les fonctions pluriharmoniques sur une variété algébrique compacte avec des singularités douces.

Classification : 32J18
Keywords: Residue, divisor, meromorphic 1-form, pluriharmonic functions
Mots-clés : Résidus, diviseur, 1-forme méromorphe, Fonction pluri harmonique

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