Real Kaehler submanifolds in codimension up to four

TL;DR

This paper classifies high-dimensional Kaehler submanifolds in Euclidean space of codimension up to four, showing they are either holomorphic or compositions involving well-understood non-holomorphic Kaehler submanifolds.

Contribution

It provides a classification of Kaehler submanifolds with complex rank at least five in codimension up to four, revealing their structure as either holomorphic or composed with known submanifolds.

Findings

Submanifolds are either holomorphic or compositions of holomorphic and non-holomorphic submanifolds.

The classification applies to manifolds with complex rank at least five.

Minimality is preserved under the composition with the submanifold $F$.

Abstract

Let $f\colon M^{2n}\to\mathbb{R}^{2n+4}$ be an isometric immersion of a Kaehler manifold of complex dimension $n\geq 5$ into Euclidean space with complex rank at least $5$ everywhere. Our main result is that, along each connected component of an open dense subset of $M^{2n}$, either $f$ is holomorphic in $\mathbb{R}^{2n+4}\cong\mathbb{C}^{n+2}$ or it is in a unique way a composition $f=F\circ h$ of isometric immersions. In the latter case, we have that $h\colon M^{2n}\to N^{2n+2}$ is holomorphic and $F\colon N^{2n+2}\to\mathbb{R}^{2n+4}$ belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifold in codimension two. Moreover, the submanifold $F$ is minimal if and only if $f$ is minimal.