I will discuss the subject of real Kaehler submanifolds, that is, isometric immersions \(f\colon M^{2n}\to\mathbb{R}^{2n+p}\) of a Kaehler manifold \((M^{2n},J)\) of complex dimension \(n\geq 2\) into Euclidean space with codimension \(p\). In particular, I will present a recent result that shows that for codimension \(2p\leq 2n-1\) generic rank conditions on the second fundamental form of \(f\) imply that the submanifold has to be minimal. In fact, for codimension \(p\leq 11\) we have a stronger conclusion, namely, that \(f\) must be holomorphic with respect to some complex structure in the ambient space.

This is joint work with A. de Carvalho and M. Dajczer.