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

TL;DR

This paper explores how K"ahlerian Killing spinors on $ ext{CP}^2$ can characterize the isometric immersion of surfaces, especially Lagrangian or complex ones, into the complex projective plane.

Contribution

It introduces a spinorial characterization of surfaces immersed in $ ext{CP}^2$ using K"ahlerian Killing spinors, linking spinor fields to geometric properties.

Findings

K"ahlerian Killing spinors characterize Lagrangian surfaces.

Spinor restrictions identify complex immersed surfaces.

New spinorial criteria for surface immersions in $ ext{CP}^2$.

Abstract

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.