Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry

DOI10.2140/GT.2020.24.1907zbMATH Open1479.32011arXiv1810.04661OpenAlexW3102878547WikidataQ125116283 ScholiaQ125116283MaRDI QIDQ2211891

Tamás Darvas, Hoang Chinh Lu

Publication date: 17 November 2020

Published in: Geometry \& Topology (Search for Journal in Brave)

Abstract: We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature K"ahler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays, and the uniform convexity properties of the space of K"ahler metrics.

Full work available at URL: https://arxiv.org/abs/1810.04661

zbMATH Keywords

stabilityKähler metricsMabuchi rays

Mathematics Subject Classification ID

Global differential geometry of Hermitian and Kählerian manifolds (53C55) Plurisubharmonic functions and generalizations (32U05) Notions of stability for complex manifolds (32Q26)

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Related Items (23)

The isometries of the space of Kähler metrics ⋮ Geodesic rays and stability in the lowercase cscK problem ⋮ Extremizers of the \(J\) functional with respect to the \(d_1\) metric ⋮ Geodesic distance and Monge-Ampére measures on contact sets ⋮ Extremal metrics on toric manifolds and homogeneous toric bundles ⋮ Mabuchi geometry of big cohomology classes ⋮ Geometric flow, multiplier ideal sheaves and optimal destabilizer for a Fano manifold ⋮ Mabuchi's soliton metric and relative D-stability ⋮ Extremal Kähler metrics of toric manifolds ⋮ A relative Yau-Tian-Donaldson conjecture and stability thresholds ⋮ The space of finite-energy metrics over a degeneration of complex manifolds ⋮ Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity ⋮ Geometric quantization of coupled Kähler-Einstein metrics ⋮ On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows ⋮ Geometric pluripotential theory on Kähler manifolds ⋮ Twisted Kähler-Einstein metrics in big classes ⋮ Sharp stability estimate for the geodesic ray transform ⋮ Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations ⋮ Valuative invariants with higher moments ⋮ Optimal asymptotic of the \(J\) functional with respect to the \(d_1\) metric ⋮ The closures of test configurations and algebraic singularity types ⋮ Comparison of Monge–Ampère capacities ⋮ Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

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