##
**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.

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.

Reviewer: G. K. Sankaran (Bath)

### MSC:

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |

14E05 | Rational and birational maps |

14D99 | Families, fibrations in algebraic geometry |

### Keywords:

singular Hermitian line bundles; moving intersection numbers; numerically trivial foliations### References:

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