Geodesic rays and stability in the cscK problem

Abstract

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archimedean entropy functional. As further applications, we show that a uniform K-stability condition for model filtrations and the ${\mathcal{J}}^{K_X}$-stability are both sufficient conditions for the existence of cscK metrics. The first condition is also conjectured to be necessary. Our arguments also produce a different proof of the toric uniform version of YTD conjecture for all polarized toric manifolds. Another result proved here is that the Mabuchi slope of a geodesic ray associated to a test configuration is equal to the non-Archimedean Mabuchi invariant.

44 pages. Updated references. Accepted by Ann. Sci. Éc. Norm. Supér

References in corpus (3)

• On the constant scalar curvature Kähler metrics, general automorphism group
• On the constant scalar curvature Kähler metrics, apriori estimates
• The closures of test configurations and algebraic singularity types

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• Uniform K-stability of polarized spherical varieties
• The closures of test configurations and algebraic singularity types
• Continuity of delta invariants and twisted Kähler--Einstein metrics
• Openness of uniformly valuative stability on the Kähler cone of projective manifolds
• Openness of uniform K-stability in the Kähler cone
• An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons
• Entropies in $\mu$-framework of canonical metrics and K-stability, I -- Archimedean aspect: Perelman's W-entropy and $\mu$-cscK metrics
• Analytical approximations and Monge-Ampère masses of plurisubharmonic singularities
• Emergent complex geometry
• A decomposition formula for J-stability and its applications
• Valuative invariants with higher moments