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**Classification of turbulent foliations.
(Classification des feuilletages turbulents.)**
*(French)*
Zbl 1074.32012

A compact complex 2-surface equipped with a holomorphic projection onto a compact Riemann surface is said to be an elliptic fibration if all its fibers are nonsingular elliptic curves except for a finite number of singular fibers. A singular one-dimensional holomorphic foliation on the same fibered surface is said to be turbulent, if it is transversal to a generic fiber. The turbulent foliations were introduced by M. Brunella [Birational geometry of foliations (2000; Zbl 1073.14022)].

The paper under review studies the space of turbulent foliations with respect to a fixed elliptic fibration. These foliations are split into natural classes. The foliations from one and the same class have the same tangency divisor with the elliptic fibration (which is an integer nonnegative linear combination of a finite number of irreducible components of fibers). The authors prove that each natural class of turbulent foliations admits a natural structure of a complex manifold. The authors also calculate its dimension. The turbulent foliations from a given class are parametrized by a complement of a complex linear space to a finite union of subvarieties. The parametrization is explicit. The authors construct appropriate holomorphic line bundle over the elliptic fibration base (this bundle depends on the prescribed tangency divisor). To each holomorphic section of this bundle they associate a singular holomorphic foliation. They show that each turbulent foliation with the prescribed tangency divisor is realized in this way by a unique section, and the latter sections form a complement to a finite union of subvarieties. The dimension of the space of holomorphic sections is calculated by using the Riemann-Roch theorem.

The paper under review studies the space of turbulent foliations with respect to a fixed elliptic fibration. These foliations are split into natural classes. The foliations from one and the same class have the same tangency divisor with the elliptic fibration (which is an integer nonnegative linear combination of a finite number of irreducible components of fibers). The authors prove that each natural class of turbulent foliations admits a natural structure of a complex manifold. The authors also calculate its dimension. The turbulent foliations from a given class are parametrized by a complement of a complex linear space to a finite union of subvarieties. The parametrization is explicit. The authors construct appropriate holomorphic line bundle over the elliptic fibration base (this bundle depends on the prescribed tangency divisor). To each holomorphic section of this bundle they associate a singular holomorphic foliation. They show that each turbulent foliation with the prescribed tangency divisor is realized in this way by a unique section, and the latter sections form a complement to a finite union of subvarieties. The dimension of the space of holomorphic sections is calculated by using the Riemann-Roch theorem.

Reviewer: Alexey A. Glutsyuk (Lyon)

### MSC:

32S65 | Singularities of holomorphic vector fields and foliations |

37F75 | Dynamical aspects of holomorphic foliations and vector fields |

### Citations:

Zbl 1073.14022
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\textit{I. Pan} and \textit{M. Sebastiani}, Ann. Fac. Sci. Toulouse, Math. (6) 12, No. 3, 395--413 (2003; Zbl 1074.32012)

### References:

[1] | Barth, W. , Peters, C., Van De Ven, A. - Compact Complex Surfaces, Springer Verlag, ( 1984). · Zbl 0718.14023 |

[2] | Brunella, M. - Feuilletages holomorphes sur les surfaces complexes compactes, Ann. Norm. Sup., 4a sér., t. 30, p. 569-594 (1997). · Zbl 0893.32019 |

[3] | Brunella, M. - Birational Geometry of Surfaces , First Latin-Americam Congress, IMPA, (2000). · Zbl 1073.14022 |

[4] | Pan, I. , Sebastiani, M. - Feuilletages tourbillonnés sur les fibrés principaux elliptiques, pré-publication. |

[5] | Gomez-Mont, X. - Universal families of foliations by curves , Astérisque, 150-151, 109-129 (1987). · Zbl 0641.32014 |

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