  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.

Differential Geometry