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Rational Curves on Foliated Varieties

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Foliation Theory in Algebraic Geometry

Part of the book series: Simons Symposia ((SISY))

Abstract

The article refines and generalises the study of deformations of a morphism along a foliation begun by Y. Miyaoka, [Mi2]. The key ingredients are the algebrisation of the graphic neighbourhood, see Fact 3.3.1, which reduces the problem from the transcendental to the algebraic, and a p-adic variation of Mori’s bend and break in order to overcome the “naive failure”, see Remark 3.2.3, of the method in the required generality. Qualitatively the results are optimal for foliations of all ranks in all dimensions, and are quantitatively optimal for foliations by curves, for which the further precision of a cone theorem is provided.

An erratum to this chapter can be found at 10.1007/978-3-319-24460-0_8

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Notes

  1. 1.

    Concomitantly with the preparation of the original pre-print, c. may 2000, J.-B. Bost, [Bo], used what may be considered an arithmetic version of this trick which independently led him to discover the geometric variant, and its higher dimensional generalisations à la § 2.1.

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Author information

Authors and Affiliations

  1. Warren Weaver Hall, Office 602, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012-1185, USA

    Fedor Bogomolov

  2. National Research University Higher School of Economics, Russian Federation, AG Laboratory, HSE,, 7 Vavilova str., Moscow, 117312, Russia

    Fedor Bogomolov

  3. Department of Mathematics, University of Rome Tor Vergata, Sogene Building, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Michael McQuillan

Authors
  1. Fedor Bogomolov
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  2. Michael McQuillan
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Corresponding author

Correspondence to Michael McQuillan .

Editor information

Editors and Affiliations

  1. Mathematics Department, Imperial College of London, London, UK

    Paolo Cascini

  2. Department of Mathematics, UC San Diego, La Jolla, CA, USA

    James McKernan

  3. IMPA, Rio de Janeiro, Brazil

    Jorge Vitório Pereira

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Bogomolov, F., McQuillan, M. (2016). Rational Curves on Foliated Varieties. In: Cascini, P., McKernan, J., Pereira, J.V. (eds) Foliation Theory in Algebraic Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-24460-0_2

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