Numerically trivial foliations

Eckl, Thomas

Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 887-938.

Étant donnée une métrique hermitienne singulière positive d’un fibré en droites sur une variété complexe kählerienne, nous construisons un feuilletage singulier satisfaisant certaines analogies analytiques des conditions numériques. Ce feuilletage raffine la fibration numériquement triviale de Tsuji et la fibration d’Iitaka. Utilisant des métriques hermitiennes singulières presque positives avec des singularités analytiques sur un fibré en droites pseudoeffectif, on construit un feuilletage raffinant la fibration nef. Si les singularités du feuilletage sont des points isolés, la codimension des feuilles est une limite supérieure pour la dimension numérique du fibré en droites, et le feuilletage donne une interprétation géométrique pour la déviation des dimensions nef et Kodaira-Iitaka. Plusieurs exemples de surfaces sont discutés, et ${\mathbb{P}}^2$ éclaté en 9 points donne un contre-exemple à l’égalité de la dimension numérique et de la codimension des feuilles.

Given a positive singular hermitian metric of a pseudoeffective line bundle on a complex Kähler manifold, a singular foliation is constructed satisfying certain analytic analogues of numerical conditions. This foliation refines Tsuji’s numerically trivial fibration and the Iitaka fibration. Using almost positive singular hermitian metrics with analytic singularities on a pseudo-effective line bundle , a foliation is constructed refining the nef fibration. If the singularities of the foliation are isolated points, the codimension of the leaves is an upper bound to the numerical dimension of the line bundle, and the foliation can be interpreted as a geometric reason for the deviation of nef and Kodaira-Iitaka dimensions. Several surface examples are studied in more details, ${\mathbb{P}}^2$ blown up in 9 points giving a counter example to equality of numerical dimension and codimension of the leaves.

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