Lines in the space of K\"ahler metrics

TL;DR

This paper develops a new correspondence for weak geodesic lines in the space of Kähler metrics, constructs diverse examples on projective manifolds, and explores geometric properties including smoothness and Euclid's postulate.

Contribution

It introduces a Ross-Witt Nyström correspondence for weak geodesics, constructs non-holomorphic generated geodesics, and examines geometric properties of the space of Kähler metrics.

Findings

Constructed weak geodesic lines not generated by holomorphic vector fields.

Disproved a folklore conjecture regarding geodesic generation.

Identified smooth geodesic lines and analyzed Euclid's fifth postulate in this context.

Abstract

We establish a Ross-Witt Nystr\"om correspondence for weak geodesic lines in the (completed) space of K\"ahler metrics. We construct a wide range of weak geodesic lines on arbitrary projective K\"ahler manifolds that are not generated by holomorphic vector fields, in the process disproving a folklore conjecture popularized by Berndtsson. Remarkably, some of these weak geodesic lines turn out to be smooth. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of K\"ahler metrics.