Lines in the space of Kähler metrics

2025/07/23 by Tamás Darvas, Darvas, Tamás, Nicholas McCleerey +1 · 1 citation
Mathematics · Physics and Astronomy · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Advanced Differential Geometry Research

paper · pdf · doi:10.48550/arxiv.2507.17375

Abstract

We establish a Ross-Witt Nyström correspondence for weak geodesic lines in the (completed) space of Kähler metrics. We construct a wide range of weak geodesic lines on arbitrary projective Kähler 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 Kähler metrics.

Citations

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Cited by

• Filtrations and asymptotic geometry of non-Archimedean norms on section rings

Related

• On the Geodesics in the space of Kähler metrics with prescribed singularities
• Space of Kähler metrics (IV)--On the lower bound of the K-energy
• Long geodesics in the space of Kähler metrics
• The Space of Kähler metrics (II)
• Riemannian geometry in infinite dimensional spaces