Geodesic rays and stability in the lowercase cscK problem. (English. French summary) Zbl 1508.58004
Author’s 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 \(\mathcal{I}^{K_X}\)-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.
Reviewer: Radu Precup (Cluj-Napoca)
MSC:
| 58E11 | Critical metrics |
| 26E30 | Non-Archimedean analysis |
| 32Q15 | Kähler manifolds |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
Keywords:
constant scalar curvature Kähler (cscK) metric; Yau-Tian-Donaldson conjecture; geodesic rays; K-stabilityReferences:
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