The CR geometry of weighted extremal Kahler and Sasaki metrics

TL;DR

This paper reveals a deep connection between conformally Einstein--Maxwell Kahler 4-manifolds and extremal Kahler 4-manifolds with non-zero scalar curvature, extending to higher dimensions via weighted extremal Kahler metrics, and explores implications for Sasaki geometry.

Contribution

It establishes a novel equivalence between certain Kahler and Sasaki metrics and extends the theory to higher dimensions using weighted extremal Kahler metrics.

Findings

Equivalence between conformally Einstein--Maxwell Kahler and extremal Kahler 4-manifolds.

Extension of the correspondence to higher dimensions with weighted extremal Kahler metrics.

New results on existence and non-existence of extremal Sasaki metrics.

Abstract

We establish an equivalence between conformally Einstein--Maxwell Kahler 4-manifolds (recently studied in many works) and extremal Kahler 4-manifolds (in the sense of Calabi) with nowhere vanishing scalar curvature. The corresponding pairs of Kahler metrics arise as transversal Kahler structures of Sasaki metrics compatible with the same CR structure and having commuting Sasaki--Reeb vector fields. This correspondence extends to higher dimensions using the notion of a weighted extremal Kahler metric, illuminating and uniting several explicit constructions in Kahler and Sasaki geometry. It also leads to new existence and non-existence results for extremal Sasaki metrics, suggesting a link between notions of relative weighted K-stability for a polarized variety, and relative K-stability of the Kahler cone corresponding to a Sasaki polarization.