Real Kaehler Submanifolds in Codimension up to Four

Let f W M2n ! R2nC4 be an isometric immersion of a Kaehler manifold of complex dimension n ≥ 5 into Euclidean space with complex rank at least 5 everywhere. Our main result is that, along each connected component of an open dense subset of M2n, either f is holomorphic in R2nC4 Š CnC2, or it is in a unique way a composition f D F ı h of isometric immersions. In the latter case, we have that hW M2n ! N2nC2 is holomorphic and F W N2nC2 ! R2nC4 belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifolds in codimension two. Moreover, the submanifold F is minimal if and only if f is minimal.