On Kähler metrisability of two-dimensional complex projective structures

We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Furthermore we show that the (pseudo-)Kähler 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 C3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}^3$$\end{document} 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.