Holomorphicity of real Kaehler submanifolds

Let $f\text{\hspace{0.17em}⁣}:M^{2n}\to {\mathbb{R}}^{2n+p}$ denote an isometric immersion of a Kaehler manifold of complex dimension $n\ge 2$ into Euclidean space with codimension $p$. If $2p\le 2n-1$, 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 $p\le 11$ we prove that $f$ must be holomorphic with respect to some complex structure in the ambient space.