Lines in the space of K\"ahler metrics
Differential Geometry
2026-02-16 v2 Complex Variables
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.
Keywords
Cite
@article{arxiv.2507.17375,
title = {Lines in the space of K\"ahler metrics},
author = {Tamás Darvas and Nicholas McCleerey},
journal= {arXiv preprint arXiv:2507.17375},
year = {2026}
}