  The complex projective space $\mathbb C P^2$ of complex dimension $2$ has a
Spin$^c$ structure carrying K\"ahlerian Killing spinors. The restriction of one
of these K\"ahlerian Killing spinors to a surface $M^2$ characterizes the
isometric immersion of $M^2$ into $\mathbb C P^2$ if the immersion is either
Lagrangian or complex.

Differential Geometry