Numerically trivial foliations

Thomas Eckl[1]

  • [1] Universität Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln (Allemagne)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 4, page 887-938
  • ISSN: 0373-0956

Abstract

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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, ℙ 2 blown up in 9 points giving a counter example to equality of numerical dimension and codimension of the leaves.

How to cite

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Eckl, Thomas. "Numerically trivial foliations." Annales de l’institut Fourier 54.4 (2004): 887-938. <http://eudml.org/doc/116137>.

@article{Eckl2004,
abstract = {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.},
affiliation = {Universität Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln (Allemagne)},
author = {Eckl, Thomas},
journal = {Annales de l’institut Fourier},
keywords = {singular hermitian line bundles; moving intersection numbers; numerically trivial foliations; singular Hermitian line bundles},
language = {eng},
number = {4},
pages = {887-938},
publisher = {Association des Annales de l'Institut Fourier},
title = {Numerically trivial foliations},
url = {http://eudml.org/doc/116137},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Eckl, Thomas
TI - Numerically trivial foliations
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 4
SP - 887
EP - 938
AB - 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.
LA - eng
KW - singular hermitian line bundles; moving intersection numbers; numerically trivial foliations; singular Hermitian line bundles
UR - http://eudml.org/doc/116137
ER -

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