On K\"ahler metrisability of two-dimensional complex projective structures

TL;DR

This paper investigates conditions under which complex projective structures on surfaces are compatible with (pseudo-)K"ahler metrics, establishing a correspondence with certain hermitian forms and deriving related curvature invariants.

Contribution

It provides necessary conditions for K"ahler metrisability of complex projective structures and characterizes compatible metrics via hermitian forms, using Cartan geometry techniques.

Findings

Necessary conditions for K"ahler metrisability derived.

One-to-one correspondence between compatible metrics and hermitian forms of rank ≥ 2.

Explicit expressions for complex projective Weyl and Liouville curvature obtained.

Abstract

We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)K\"ahler metric. Furthermore we show that the (pseudo-)K\"ahler metrics defined on some domain in the projective plane which are compatible with the standard complex projective structure are in one-to-one correspondence with the hermitian forms on $\mathbb{C}^3$ whose rank is at least two. This is achieved by prolonging the relevant finite-type first order linear differential system to closed form. Along the way we derive the complex projective Weyl and Liouville curvature using the language of Cartan geometries.