Numerically trivial foliations. (English) Zbl 1071.32018

There are at least three ways of attaching a fibration on a projective complex manifold \(X\) to a nef (or otherwise suitably positive) line bundle \(L\). This paper compares them and goes a considerable way towards explaining explaining how and why they differ.
The nef fibration, defined by T. Bauer et al. [Bauer, I. (ed.) et al., Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer. 27–36 (2002; Zbl 1054.14019)], has as its fibres curves \(C\) with \(L. C=0\). The dimension of the base is called the nef dimension (denoted \(n(L)\)): one has \(n(L)\geq \nu(L)\), the numerical dimension of \(L\). Secondly, the author [J. Algebr. Geom. 13, No. 4, 617–639 (2004; Zbl 1065.14009)], following ideas of Tsuji, constructed a fibration associated to a positive singular hermitian metric \(h\) on \(L\), for which the fibres satisfy \((L,h). C=0\). The intersection number \((L,h). C\) used there is the one defined in that context by Tsuji in terms of multiplier ideal sheaves. Thirdly, the classical Iitaka fibration can also be defined in a similar way, using intersection numbers defined via the asymptotic ideal sheaf: this was done by S. Takayama [Trans. Am. Math. Soc. 355, No. 1, 37–47 (2003; Zbl 1055.14011)].
The unifying idea introduced here is to construct numerically trivial foliations with respect to a positive closed \((1,1)\)-current \(T\) (for example, the curvature current of a suitable Hermitian metric on \(L\)). The main technical result in this direction is that for such \(T\) there exists a maximal foliation with leaves numerically trivial with respect to \(T\): the leaves of any other such foliation are contained in the leaves of this one.
Tsuji’s fibration and the Iitaka fibration then emerge from this construction, as the maximal fibration among those whose fibres are contained in the leaves of the maximal numerically trivial fibration with respect to the curvature of the Hermitian metric. The nef fibration does not quite fit into this picture. The following variant, inspired by the idea of moving intersection numbers, is needed: we have \[ L. C=\lim_{\varepsilon\to 0^+} \sup_T\int_{C\setminus {\operatorname{Sing}}T} (T+\varepsilon\omega) \] where \(\omega\) is a Kähler form and \(T\geq -\varepsilon \omega\) is a closed current representing \(c_1(L)\). So \(L. C=0\) if and only if \[ \lim_{\varepsilon\to 0^+} \sup_T\int_{\Delta\setminus {\operatorname{Sing}}T} (T+\varepsilon\omega)=0 \] for every disc \(\Delta\subset C\): in other words, numerical triviality is a local property. This allows the author to show that again there is a maximal numerically trivial foliation.
Using this approach the author shows that the Iitaka dimension \(\kappa(L)<n(L)\) if the nef foliation is not a fibration, and he asks whether the converse is true. The numerical dimension is bounded above by the codimension of the leaves, provided the foliation has isolated singularities: it is not yet clear how, if at all, this restriction may be removed.
Finally, some nef fibrations on surfaces are described directly in terms of foliations.


32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14E05 Rational and birational maps
14D99 Families, fibrations in algebraic geometry
Full Text: DOI arXiv Numdam EuDML


[1] A reduction map for nef line bundles, Analytic and Algebraic Methods in Complex Geometry. Konferenzbericht der Konferenz zu Ehren von Hans Grauert, Goettingen (2000) · Zbl 1054.14019
[2] Complex Tori, 177 (1999) · Zbl 0945.14027
[3] Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., 128, 207-302 (1997) · Zbl 0896.14006
[4] Inégalités de Morse et variétés de Moishezon (1995)
[5] On the volume of a line bundle (2001) · Zbl 1101.14008
[6] Cônes positifs des variétés complexes compactes (2002)
[7] Higher dimensional Zariski decompositions (2002)
[8] Birational geometry of fibrations., First Latin American Congress of Mathematicians, IMPA, July 31-August 4, 2000 (2000)
[9] The Dirichlet Problem for a complex Monge-Ampère equation, Invent. Math., 37, 1-44 (1976) · Zbl 0315.31007
[10] A Subadditivity Property of Multiplier Ideals, Michigan Math. J., 48, 137-156 (2000) · Zbl 1077.14516
[11] Multiplier ideal sheaves and analytic methods in algebraic geometry, School on Vanishing theorems and effective results in Algebraic Geometry, ICTP Trieste (2000) · Zbl 1102.14300
[12] Private communication (2002)
[13] Estimations \(L^2\) pour l’opérateur \(\overline\partial\) d’un fibré vectoriel holomorphe semi-positif au dessus d’une variété kählerienne complète, Ann. Sci. ENS, 15, 457-511 (1982) · Zbl 0507.32021
[14] Regularization of closed positive currents and Intersection theory, J. Alg. Geom., 1, 361-409 (1992) · Zbl 0777.32016
[15] Treatise on Analysis II (1970) · Zbl 0202.04901
[16] Pseudo-effective line bundles on compact kähler manifolds, Int. J. Math., 12, 6, 689-741 (2001) · Zbl 1111.32302
[17] Compact complex manifolds with numerically effective tangent bundles, J. Alg. Geom., 3, 295-345 (1994) · Zbl 0827.14027
[18] Kähler manifolds with semipositive anticanonical bundle, Comp. Math., 101, 217-224 (1996) · Zbl 1008.32008
[19] Tsuji’s Numerical Trivial Fibrations (2002) · Zbl 1065.14009
[20] Algebraic surface and holomorphic vector bundles (1998) · Zbl 0902.14029
[21] Approximating Zariski decomposition of big line bundles, Kodai Math. J., 17, 1-3 (1994) · Zbl 0814.14006
[22] Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math., 79, 109-326 (1964) · Zbl 0122.38603
[23] Algebraic Geometry, 76 (1982) · Zbl 0491.14006
[24] Deformations of canonical singularities, J. Amer. Math. Soc., 12, 85-92 (1999) · Zbl 0906.14001
[25] Multiplier ideals for algebraic geometers (2000)
[26] Fonctions Plurisousharmonique et Formes Différentielles Positives (1968) · Zbl 0195.11603
[27] Opérateur de Monge-Ampère et Tranchage des Courants Positifs Fermés, J. Geom. Analysis, 10, 1, 139-168 (2000) · Zbl 1005.32023
[28] Deformations of a morphism along a foliation and applications, Proc. Symp. Pure Math., 46, 1, 245-268 (1987) · Zbl 0659.14008
[29] Vector bundles on complex projective spaces, 3 (1980) · Zbl 0438.32016
[30] Iitaka’s fibration via multiplier ideals, Trans. AMS, 355, 37-47 (2002) · Zbl 1055.14011
[31] Numerically trivial fibrations (2000)
[32] Existence and applications of the Analytic Zariski Decomposition, Analysis and geometry in several complex variables, 253-271 (1999) · Zbl 0965.32022
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