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COMPLEX SUBMANIFOLDS IN REAL HYPERSURFACES

  • Han, Chong-Kyu ;
  • Tomassini, Giuseppe
  • Received : 2009.01.05
  • Published : 2010.09.01

Abstract

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

Keywords

extension of holomorphic functions;real hypersurfaces in complex manifolds;complex submanifolds;Levi-form;Pfaffian system;generalized Frobenius theorem

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Cited by

  1. Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds vol.22, pp.2, 2012, https://doi.org/10.1007/s12220-010-9208-2

Acknowledgement

Grant : Geometric Properties of Real and Complex Manifolds

Supported by : Korea Research Foundation, MURST