COMPLEX SUBMANIFOLDS IN REAL HYPERSURFACES

Title & Authors
COMPLEX SUBMANIFOLDS IN REAL HYPERSURFACES
Han, Chong-Kyu; Tomassini, Giuseppe;

Abstract
Let M be a $\small{C^{\infty}}$ real hypersurface in $\small{\mathbb{C}^{n+1}}$, $\small{n\;{\geq}\;1}$, locally given as the zero locus of a $\small{C^{\infty}}$ real valued function r that is defined on a neighborhood of the reference point $\small{P\;{\in}\;M}$. 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 $\small{i{\partial}r}$ 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;
Language
English
Cited by
1.
Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds, Journal of Geometric Analysis, 2012, 22, 2, 561
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