Holomorphicity of real Kaehler submanifolds

Let f : M 2n → R 2n+p denote an isometric immersion of a Kaehler manifold of complex dimension n ≥ 2 into Euclidean space with codimension p.If 2p ≤ 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 ≤ 11 we prove that f must be holomorphic with respect to some complex structure in the ambient space.