Real Kähler submanifolds in codimension 6

Abstract

We show that a real Kähler submanifold in codimension 6 6 is essentially a holomorphic submanifold of another real Kähler submanifold in lower codimension if the second fundamental form is not sufficiently degenerated. We also give a shorter proof of this result when the real Kähler submanifold is minimal, using recent results about isometric rigidity.