The c-projective symmetry algebras of Kähler surfaces

TL;DR

This work determines the full c-projective symmetry algebras for Kähler surfaces with essential c-projective vector fields by first classifying metrics with constant holomorphic sectional curvature (HSC) and then computing algebras for non-constant HSC metrics, organized into Liouville, complex, and degenerate types. It employs mobility theory (with $D([g])$) and Sinjukov equations to constrain the possible c-projective fields, and leverages a unifying lifting mechanism from a 2D base metric $h$ to the full 4D Kähler metric. The paper provides explicit algebraic descriptions and generators for each metric family (L1–L4, C1–C4, D1–D3), with dimensions ranging from 2 to 5, and shows that constant-HSC cases share the same c-projective algebra as the Fubini-Study metric. The results clarify how c-projective algebras behave under c-projective transformations and set the stage for extending the analysis to higher dimensions and to isometry classifications in follow-up work.

Abstract

Let $M$ be a Kähler manifold with complex structure $J$ and Kähler 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 Kähler surfaces with essential (i.e., non-affine) c-projective vector fields are computed.

Figures (1)

• Figure 1: The gnomonic (central) projection maps great circles to lines.

Theorems & Definitions (26)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Example 1: Fubini-Study metric on $\mathbb{C}\mathbb{P}^2$
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1: MD1978MS1983BMMR2015