Abstract
The goal of this note is to exhibit the integrability properties (in the sense of the Frobenius theorem) of holomorphic p-forms with values in certain line bundles with semi-negative curvature on a compact Kähler manifold. There are in fact very strong restrictions, both on the holomorphic form and on the curvature of the semi-negative line bundle. In particular, these observations provide interesting information on the structure of projective manifolds which admit a contact structure: either they are Fano manifolds or, thanks to results of Kebekus-Peternell-Sommese-Wisniewski, they are biholomorphic to the projectivization of the cotangent bundle of another suitable projective manifold.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Beauville, A., Fano contact manifolds and nilpotent orbits. Comm. Math. Helv. 73(4) (1998), 566–583.
Beauville, A., Riemannian holonomy and algebraic geometry, Duke/alggeom preprint 9902110, 1999.
Boothby, W., Homogeneous complex contact manifolds, Proc. Symp. Pure. Math. (Differential Geometry) 3 (1961), 144–154.
Bedford, E., Taylor, B. A., The Dirichlet problem for a complex Monge-Ampère equation. Invent. Math. 37 (1976), 1–44.
Bedford, E., Taylor, B. A., A new capacity for plurisubharmonic functions. Acta Math. 149 (1982), 1–41.
Demailly, J.-P., Singular Hermitian metrics on positive line bundles., in: Hulek K., Peternell T., Schneider M., Schreyer F. (Eds.), Proceedings of the Bayreuth Conference “complex Algebraic Varieties,” April 2–6, 1990, Lecture Notes in Math. 1507 (1992), Springer-Verlag.
Demailly, J.-P., Regularization of closed positive currents and intersection theory, J. Alg. Geom. 1 (1992), 361–409.
Druel, S., Contact structures on Algebraic 5-dimensional manifolds, C.R. Acad. Sci. Paris 327 (1998), 365–368.
Kebekus, S., Peternell, Th., Sommese, A. J., Wiśniewski, J. A., Projective contact manifolds, 1999, Invent. Math., to appear.
Wolf, J., Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech. 14 (1965), 1033–1047.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Demailly, JP. (2002). On the Frobenius Integrability of Certain Holomorphic p-Forms. In: Bauer, I., Catanese, F., Peternell, T., Kawamata, Y., Siu, YT. (eds) Complex Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56202-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-56202-0_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62790-3
Online ISBN: 978-3-642-56202-0
eBook Packages: Springer Book Archive