(Pseudo-)K\"ahler-Einstein geometries

TL;DR

This paper systematically studies Einstein complex geometries derived from logarithmic Kähler potentials, exploring how different coordinate choices affect the metric signature and the domain of definition.

Contribution

It introduces a classification of Einstein complex geometries based on logarithmic Kähler potentials and analyzes their domain and signature properties.

Findings

Classification of geometries into direct, inverted, and hybrid types.

Relation between coordinate choice and metric signature.

Determination of the maximum domain of definition for each type.

Abstract

Solutions to vacuum Einstein field equations with cosmological constant, such as the de Sitter space and the anti-de Sitter space, are basic in different cosmological and theoretical developments. It is also well known that complex structures admit metrics of this type. The most famous example is the complex projective space endowed with the Fubini-Study metric. In this work, we perform a systematic study of Einstein complex geometries derived from a logarithmic K\"ahler potential. Depending on the different contribution to the argument of such logarithmic term, we shall distinguish among direct, inverted and hybrid coordinates. They are directly related to the signature of the metric and determine the maximum domain of the complex space where the geometry can be defined.