Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Rationally connected foliations after Bogomolov and McQuillan


Authors: Stefan Kebekus, Luis Solá Conde and Matei Toma
Journal: J. Algebraic Geom. 16 (2007), 65-81
DOI: https://doi.org/10.1090/S1056-3911-06-00435-8
Published electronically: May 25, 2006
MathSciNet review: 2257320
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Abstract | References | Additional Information

Abstract: This paper is concerned with sufficient criteria to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov and McQuillan and using the recent work of Graber, Harris, and Starr, we give a clean, short and simple proof of previous results. Apart from a new vanishing theorem for vector bundles in positive characteristic, our proof employs only standard techniques of Mori theory and does not make any reference to the more involved properties of foliations in characteristic $ p$.

We also give a new sufficient condition to ensure that all leaves are algebraic.

The results are then applied to show that $ \mathbb{Q}$-Fano varieties with unstable tangent bundles always admit a sequence of partial rational quotients naturally associated to the Harder-Narasimhan filtration.


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Additional Information

Stefan Kebekus
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
Email: stefan.kebekus@math.uni-koeln.de

Luis Solá Conde
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
Email: lsola@math.uni-koeln.de

Matei Toma
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany and Mathematical Institute of the Romanian Academy, Bucharest
Email: matei@math.uni-koeln.de

DOI: https://doi.org/10.1090/S1056-3911-06-00435-8
Received by editor(s): May 13, 2005
Received by editor(s) in revised form: September 3, 2005, and September 29, 2005
Published electronically: May 25, 2006
Additional Notes: The authors thank their hosting institution, Universität zu Köln. The first two authors were supported in full or in part by the Forschungsschwerpunkt “Globale Methoden in der komplexen Analysis” of the Deutsche Forschungsgemeinschaft. A part of this paper was worked out while Stefan Kebekus visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation.

American Mathematical Society