Real Kähler Submanifolds in Codimension \(6\)

We show that a real K\"ahler submanifold in codimension \(6\) is essentially a holomorphic submanifold of another real K\"ahler 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 K\"ahler submanifold is minimal, using recent results about isometric rigidity.