Conformal Kaehler Euclidean submanifolds

Let $f\text{\hspace{0.17em}⁣}:M^{2n}\to {\mathbb{R}}^{2n+\mathrm{\ell }}$, $n\ge 5$, denote a conformal immersion into Euclidean space with codimension $\mathrm{\ell }$ of a Kaehler manifold of complex dimension $n$ and free of flat points. For codimensions $\mathrm{\ell }=1,2$ we show that such a submanifold can always be locally obtained in a rather simple way, namely, from an isometric immersion of the Kaehler manifold $M^{2n}$ into either ${\mathbb{R}}^{2n+1}$ or ${\mathbb{R}}^{2n+2}$, the latter being a class of submanifolds already extensively studied.