Algebraicity of foliations on complex projective manifolds, applications

Frédéric Campana; Frédéric Campana

doi:10.3934/era.2022063

Electronic Research Archive

2022, Volume 30, Issue 4: 1187-1208. doi: 10.3934/era.2022063

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Algebraicity of foliations on complex projective manifolds, applications

Frédéric Campana ^,

Université Lorraine, Nancy, France

Received: 22 August 2021 Revised: 17 November 2021 Accepted: 09 December 2021 Published: 14 March 2022

This is an expository text, originally intended for the ANR 'Hodgefun' workshop, twice reported, organised at Florence, villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the 'non pseudoeffectivity' of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from ^[1], which is inspired by the former results ^[2,3,4]. In order to make things as simple as possible, we present here only the projective versions of these results, although most of them can be easily extended to the logarithmic or 'orbifold' context.
- algebraicity of foliations,
- rationally connected varieties,
- movable classes,
- numerical dimension,
- pseudoeffectivity
Citation: Frédéric Campana. Algebraicity of foliations on complex projective manifolds, applications[J]. Electronic Research Archive, 2022, 30(4): 1187-1208. doi: 10.3934/era.2022063

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Abstract

This is an expository text, originally intended for the ANR 'Hodgefun' workshop, twice reported, organised at Florence, villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the 'non pseudoeffectivity' of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from ^[1], which is inspired by the former results ^[2,3,4]. In order to make things as simple as possible, we present here only the projective versions of these results, although most of them can be easily extended to the logarithmic or 'orbifold' context.

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