Holomorphicity of Real Kaehler Submanifolds

Let f (Formula Presented) denote an isometric immersion of a Kaehler manifold of complex dimension n ≥ 2 into Euclidean space with codimension p. If the codimension satisfies (Formula Presented), we show that generic rank assumptions on the second fundamental form of the submanifold imply that f has to be a minimal submanifold. In fact, if the codimension is p ≤ 11 we prove that f must be holomorphic with respect to some complex structure in the ambient space.