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Characteristic foliation on a hypersurface of general type in a projective symplectic manifold

Part of: Complex dynamical systems Surfaces and higher-dimensional varieties Curves

Published online by Cambridge University Press:  26 January 2010

Jun-Muk Hwang  and
Eckart Viehweg
Jun-Muk Hwang
Affiliation:
Korea Institute for Advanced Study, Hoegiro 87, Seoul 130-722, Korea (email: jmhwang@kias.re.kr)
Eckart Viehweg
Affiliation:
Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany (email: viehweg@uni-due.de)
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Abstract

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A foliation on a non-singular projective variety is algebraically integrable if all leaves are algebraic subvarieties. A non-singular hypersurface X in a non-singular projective variety M equipped with a symplectic form has a naturally defined foliation, called the characteristic foliation on X. We show that if X is of general type and dim M≥4, then the characteristic foliation on X cannot be algebraically integrable. This is a consequence of a more general result on Iitaka dimensions of certain invertible sheaves associated with algebraically integrable foliations by curves. The latter is proved using the positivity of direct image sheaves associated to families of curves.

Keywords

holomorphic foliationprojective symplectic variety

MSC classification

Secondary: 37F75: Holomorphic foliations and vector fields 14J40: $n$-folds ($n>4$) 14H10: Families, moduli (algebraic)
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 2 , March 2010 , pp. 497 - 506
Copyright
Copyright © Foundation Compositio Mathematica 2010

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