The c-projective symmetry algebras of K\"ahler surfaces

TL;DR

This paper classifies the algebraic structure of symmetries preserving c-projective geometry on K"ahler surfaces, focusing on those with non-trivial transformations that preserve complex structure and J-planar curves.

Contribution

It explicitly computes the c-projective symmetry algebras for K"ahler surfaces with essential c-projective vector fields, advancing understanding of their geometric symmetry structures.

Findings

Determined the structure of c-projective symmetry algebras for specific K"ahler surfaces.

Identified conditions under which these algebras are non-trivial.

Provided classifications for surfaces with essential c-projective symmetries.

Abstract

Let $M$ be a K\"ahler manifold with complex structure $J$ and K\"ahler metric $g$. A c-projective vector field is a vector field on $M$ whose flow sends $J$-planar curves to $J$-planar curves, where $J$-planar curves are analogs of what (unparametrised) geodesics are for pseudo-Riemannian manifolds (without complex structure). The c-projective symmetry algebras of K\"ahler surfaces with essential (i.e., non-affine) c-projective vector fields are computed.