Abstract
In this chapter we explain a remarkable theorem of Miyaoka [32] which asserts that a foliation whose cotangent bundle is not pseudoeffective is a foliation by rational curves. The original Miyaoka’s proof can be thought as a foliated version of Mori’s technique of construction of rational curves by deformations of morphisms in positive characteristic [33].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
A. Andreotti, Théorèmes de dépendance algébrique sur les espaces complexes pseudoconcaves. Bull. SMF 91, 1–38 (1963)
W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces (Springer, Heidelberg, 1984)
F.A. Bogomolov, M. McQuillan, Rational curves on foliated varieties. Preprint IHES (2001)
M. Brunella, Feuilletages holomorphes sur les surfaces complexes compactes. Ann. Sci. ENS 30, 569–594 (1997)
J.-P. Demailly, Regularization of closed positive currents and intersection theory. J. Algebraic Geom. 1, 361–409 (1992)
T. Ekedahl, Canonical models of surfaces of general type in positive characteristic. Publ. Math. IHES 67, 97–144 (1988)
T. Fujita, On Zariski problem. Proc. Japan Acad. Ser. A Math. Sci. no. 3, 106–110 (1979)
T. Kizuka, On the movement of the Poincaré metric with the pseudoconvex deformation of open Riemann surfaces. Ann. Acad. Sci. Fenn. 20, 327–331 (1995)
Y. Miyaoka, Deformation of a morphism along a foliation and applications, in Algebraic Geometry, Bowdoin 1985. Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, 1987), pp. 245–268
Y. Miyaoka, T. Peternell, Geometry of Higher Dimensional Algebraic Varieties (Birkhäuser, Basel, 1997)
C. Seshadri, L’opération de Cartier, in Séminaire Chevalley 1958–59, Exposé 6, ENS (Secrétariat Mathématique, Paris, 1960)
N.I. Shepherd-Barron, Miyaoka’s theorems on the generic seminegativity of T X and on the Kodaira dimension of minimal regular threefolds, in Flips and Abundance for Algebraic Threefolds. Astérisque, vol. 211 (Société Mathématique de France, Paris, 1992), pp. 103–114
M. Suzuki, Sur les intégrales premières de certains feuilletages analytiques complexes, in Fonctions de Plusieurs Variables Complexes III, ed. by F. Norguet. Lecture Notes in Mathematics, vol. 670 (Springer, Heidelberg, 1977), pp. 53–79
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Brunella, M. (2015). The Rationality Criterion. In: Birational Geometry of Foliations. IMPA Monographs, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-14310-1_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-14310-1_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-14309-5
Online ISBN: 978-3-319-14310-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)