  A beautiful solution to the problem of isometric immersions in $\mathbb{R}^n$
using spinors was found by Bayard, Lawn and Roth. However to use spinors one
must assume that the manifold carries a $\mbox{Spin}$-structure and, especially
for complex manifolds where is more natural to consider
$\mbox{Spin}^{\mathbb{C}}$-structures, this hypothesis is somewhat restrictive.
In the present work we show how the above solution can be adapted to
$\mbox{Spin}^{\mathbb{C}}$-structures.
