On Kähler metrisability of two-dimensional complex projective structures

Abstract

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 ${\mathbb{C}}^3$ 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.

17 pages, exposition improved, typos corrected, references added

References in corpus (2)

• Geodesically equivalent metrics in general relativity
• Reduction of beta-integrable 2-Segre structures

Cited by in corpus (8)

• Light cone and Weyl compatibility of conformal and projective structures
• Convex projective surfaces with compatible Weyl connection are hyperbolic
• Geodesic rigidity of conformal connections on surfaces
• GL(2)-geometry and complex structures
• Metrisability of projective surfaces and pseudo-holomorphic curves
• Projectively equivalent 2-dimensional superintegrable systems with projective symmetries
• Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature
• Weyl metrizability of 3-dimensional projective structures and CR submanifolds