Holomorphicity of real Kaehler submanifolds

TL;DR

This paper investigates conditions under which Kaehler submanifolds immersed in Euclidean space are forced to be holomorphic, showing that low codimension and certain rank conditions imply holomorphicity or minimality.

Contribution

It establishes new criteria linking codimension, second fundamental form rank, and holomorphicity of Kaehler submanifolds in Euclidean space.

Findings

For codimension p ≤ 11, submanifolds are holomorphic.

Generic rank conditions lead to minimality.

Low codimension enforces holomorphic structure.

Abstract

Let $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with codimension $p$. If $2p\leq 2n-1$, we show that generic rank conditions on the second fundamental form of the submanifold imply that $f$ has to be a minimal submanifold. In fact, for codimension $p\leq 11$ we prove that $f$ must be holomorphic with respect to some complex structure in the ambient space.