no. 4
Numerically trivial foliations
[Feuilletages numériquement triviaux]
Eckl, Thomas1
1 Universität Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln (Allemagne)
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 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, ℙ 2 blown up in 9 points giving a counter example to equality of numerical dimension and codimension of the leaves.

DOI : 10.5802/aif.2038
Classification : 32J25
Keywords: singular hermitian line bundles, moving intersection numbers, numerically trivial foliations
Mot clés : fibrés en droites hermitiens singuliers, nombres d'intersections mobiles, feuilletages numériquement triviaux

Eckl, Thomas 1

1 Universität Köln, Mathematisches Institut, Weyertal 86-90, 50931 Köln (Allemagne) 
@article{AIF_2004__54_4_887_0,
     author = {Eckl, Thomas},
     title = {Numerically trivial foliations},
     journal = {Annales de l'Institut Fourier},
     pages = {887--938},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {4},
     year = {2004},
     doi = {10.5802/aif.2038},
     zbl = {1071.32018},
     mrnumber = {2111016},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2038/}
}
TY  - JOUR
AU  - Eckl, Thomas
TI  - Numerically trivial foliations
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 887
EP  - 938
VL  - 54
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2038/
DO  - 10.5802/aif.2038
LA  - en
ID  - AIF_2004__54_4_887_0
ER  - 
%0 Journal Article
%A Eckl, Thomas
%T Numerically trivial foliations
%J Annales de l'Institut Fourier
%D 2004
%P 887-938
%V 54
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2038/
%R 10.5802/aif.2038
%G en
%F AIF_2004__54_4_887_0
Eckl, Thomas. Numerically trivial foliations. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 887-938. doi : 10.5802/aif.2038. https://aif.centre-mersenne.org/articles/10.5802/aif.2038/

[BCE+00] Th. Bauer; F. Campana; Th. Eckl; St. Kebekus; Th. Peternell; S. Rams; T. Szemberg; L. Wotzlaw A reduction map for nef line bundles, Analytic and Algebraic Methods in Complex Geometry. Konferenzbericht der Konferenz zu Ehren von Hans Grauert, Goettingen (April 2000) | Zbl

[BL99] Ch. Birkenhake; H. Lange Complex Tori, Progress in Mathematics, 177, Birkhäuser, 1999 | MR | Zbl

[BM97] E. Bierstone; P. D. Milman Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., Volume 128 (1997), pp. 207-302 | MR | Zbl

[Bon95] L. Bonavero Inégalités de Morse et variétés de Moishezon (1995) (e-print, alg-geom/9512013)

[Bou01] S. Boucksom On the volume of a line bundle (2001) (Preprint) | MR | Zbl

[Bou02a] S. Boucksom Cônes positifs des variétés complexes compactes (2002) (PhD thesis, Grenoble)

[Bou02b] S. Boucksom Higher dimensional Zariski decompositions (2002) (e-print, math.AG/0204336)

[Bru00] M. Brunella Birational geometry of fibrations., First Latin American Congress of Mathematicians, IMPA, July 31-August 4, 2000 (2000)

[BT76] E. Bedford; B.A. Taylor The Dirichlet Problem for a complex Monge-Ampère equation, Invent. Math., Volume 37 (1976), pp. 1-44 | MR | Zbl

[DEL00] J.-P. Demailly; L. Ein; R. Lazarsfeld A Subadditivity Property of Multiplier Ideals, Michigan Math. J., Volume 48 (2000), pp. 137-156 | MR | Zbl

[Dem00] J.-P. Demailly Multiplier ideal sheaves and analytic methods in algebraic geometry, School on Vanishing theorems and effective results in Algebraic Geometry, ICTP Trieste (Lecture Notes) (April 2000) | Zbl

[Dem02] J.-P. Demailly Private communication (2002)

[Dem82] J.-P. Demailly Estimations L 2 pour l'opérateur ∂ ¯ d'un fibré vectoriel holomorphe semi-positif au dessus d'une variété kählerienne complète, Ann. Sci. ENS, Volume 15 (1982), pp. 457-511 | Numdam | MR | Zbl

[Dem92] J.-P. Demailly Regularization of closed positive currents and Intersection theory, J. Alg. Geom., Volume 1 (1992), pp. 361-409 | MR | Zbl

[Die70] J. Dieudonné Treatise on Analysis II, Academic Press, 1970 | MR | Zbl

[DPS01] J.-P. Demailly; Th. Peternell; M. Schneider Pseudo-effective line bundles on compact kähler manifolds, Int. J. Math., Volume 12 (2001) no. 6, pp. 689-741 | MR | Zbl

[DPS94] J.-P. Demailly; Th. Peternell; M. Schneider Compact complex manifolds with numerically effective tangent bundles, J. Alg. Geom., Volume 3 (1994), pp. 295-345 | MR | Zbl

[DPS96] J.-P. Demailly; Th. Peternell; M. Schneider Kähler manifolds with semipositive anticanonical bundle, Comp. Math., Volume 101 (1996), pp. 217-224 | Numdam | MR | Zbl

[Eck02] Thomas Eckl Tsuji's Numerical Trivial Fibrations (2002) (e-print. To appear in J. Alg. Geom, math.AG/0202279) | Zbl

[Fri98] R. Friedman Algebraic surface and holomorphic vector bundles, Springer, 1998 | MR | Zbl

[Fuj94] T. Fujita Approximating Zariski decomposition of big line bundles, Kodai Math. J., Volume 17 (1994), pp. 1-3 | MR | Zbl

[Hir64] H. Hironaka Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math., Volume 79 (1964), pp. 109-326 | MR | Zbl

[Iit82] S. Iitaka Algebraic Geometry, Graduate Texts in Math., 76, Springer, New York, 1982 | MR | Zbl

[Kaw99] Y. Kawamata Deformations of canonical singularities, J. Amer. Math. Soc., Volume 12 (1999), pp. 85-92 | MR | Zbl

[Laz00] R. Lazarsfeld Multiplier ideals for algebraic geometers (August 2000) (preprint, http://www.math.lsa.umich.edu/~rlaz/)

[Lel68] P. Lelong Fonctions Plurisousharmonique et Formes Différentielles Positives, Gordon and Breach, London, 1968 | MR | Zbl

[ME00] H. Ben Messaoud; H. ElMir Opérateur de Monge-Ampère et Tranchage des Courants Positifs Fermés, J. Geom. Analysis, Volume 10 (2000) no. 1, pp. 139-168 | MR | Zbl

[Miy86] Y. Miyaoka Deformations of a morphism along a foliation and applications, Proc. Symp. Pure Math., Volume 46 (1987) no. 1, pp. 245-268 | MR | Zbl

[OSS80] Ch. Okonek; M. Schneider; H. Spindler Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser, 1980 | MR | Zbl

[Tak02] S. Takayama Iitaka's fibration via multiplier ideals, Trans. AMS, Volume 355 (2002), pp. 37-47 | MR | Zbl

[Tsu00] H. Tsuji Numerically trivial fibrations (2000) (Preprint)

[Tsu99] H. Tsuji; G. Komatsu Existence and applications of the Analytic Zariski Decomposition, Analysis and geometry in several complex variables (Trends in mathematics) (1999), pp. 253-271 | Zbl

Cité par Sources :