An Extension Theorem for Real Kahler Submanifolds in Codimension Four

TL;DR

This paper proves a new extension theorem for real Kahler submanifolds in codimension four, showing they can be embedded as holomorphic hypersurfaces in higher-dimensional Kahler manifolds, extending previous results.

Contribution

It generalizes the Kahler extension theorem to codimension four, providing a broader understanding of the structure of real Kahler submanifolds.

Findings

Real Kahler submanifolds of codimension 4 and rank ≥ 5 are holomorphic hypersurfaces in codimension 2 Kahler manifolds.

Extension theorem extends previous results from codimension 3 to codimension 4.

The main theorem broadens the class of submanifolds for which Kahler extensions are known.

Abstract

In this article, we prove a Kahler extension theorem for real Kahler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kahler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states that any real Kahler submanifolds of codimension 3 and rank at least 4 admits a Kahler extension.