Classification of turbulent foliations
type: Item
instance of: scholarly article
title: Classification of turbulent foliations
zbMATH Open document ID: 1074.32012
author: Iván Pan
author: Marcos Sebastiani
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2004-10-25T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73609
review text: 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 \textit{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.
Mathematics Subject Classification ID: 32S65
Mathematics Subject Classification ID: 37F75
zbMATH DE Number: 2109727
zbMATH Keywords: holomorphic foliation
zbMATH Keywords: elliptic fibration
zbMATH Keywords: turbulent foliation
zbMATH Keywords: tangency divisor
reviewed by: Alexey A. Glutsyuk
MaRDI profile type: Publication
OpenAlex ID: W2320633628
cites work: Q5749404
cites work: Feuilletages holomorphes sur les surfaces complexes compactes
cites work: Q4801049
cites work: Q3782016
DOI: 10.5802/AFST.1054
description: scientific article; zbMATH DE number 2109727