Journal of the
Korean Mathematical Society JKMS

Let $M$ be a $C^\infty$ real hypersurface in $\mathbb C^{n+1}$, $n\ge 1,$ 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 $k=1, \ldots, n$ 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

MSC numbers: 32D15, 58A10, 58A17