Complex and Lagrangian surfaces of the complex projective plane via Kählerian Killing Spinc spinors

The complex projective space CP2 of complex dimension 2 has a Spinc structure carrying Kählerian Killing spinors. The restriction of one of these Kählerian Killing spinors to a surface M2 characterizes the isometric immersion of M2 into CP2 if the immersion is either Lagrangian or complex.