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Chapitre D'ouvrage Année : 2023

The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds

Résumé

We prove the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds, that is, for projective manifolds equipped with a holomorphic action of a compact Lie group with at least one real hypersurface orbit. Contrary to what seems to be a popular belief, such manifolds do not admit extremal Kähler metrics in all Kähler classes in general. More generally, we prove that for rank one polarized spherical varieties, G-uniform K-stability is equivalent to K-stability with respect to special G-equivariant test configurations. This is furthermore encoded by a single combinatorial condition, checkable in practice. We illustrate on examples and answer along the way a question of Kanemitsu.
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Dates et versions

hal-03005597 , version 1 (14-11-2020)

Identifiants

Citer

Thibaut Delcroix. The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds. Birational Geometry, Kähler–Einstein Metrics and Degenerations, 409, Springer International Publishing, pp.205-223, 2023, Springer Proceedings in Mathematics & Statistics, ⟨10.1007/978-3-031-17859-7_10⟩. ⟨hal-03005597⟩
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