Complex and Lagrangian surfaces of the complex projective plane via K\"ahlerian Killing Spin$^c$ spinors

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.