# Real K\"ahler Submanifolds in Codimension $6$

## TL;DR

This paper investigates the structure of real K"ahler submanifolds in codimension 6, demonstrating they are closely related to holomorphic submanifolds under certain conditions, with a simplified proof for minimal cases.

## Contribution

It establishes a new geometric characterization of real K"ahler submanifolds in codimension 6 and provides a shorter proof for the minimal case leveraging recent rigidity results.

## Key findings

- Real K"ahler submanifolds in codimension 6 are essentially holomorphic submanifolds of lower codimension.
- A shorter proof is provided for the minimal case using isometric rigidity results.
- The structure depends on the degeneracy of the second fundamental form.

## Abstract

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.