Immersion in ${\mathbb{R}}^n$ by complex spinors

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 $Spin$-structure and, especially for complex manifolds where is more natural to consider $Spi{n}^{\mathbb{C}}$-structures, this hypothesis is somewhat restrictive. In the present work we show how the above solution can be adapted to $Spi{n}^{\mathbb{C}}$-structures.