The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds
Algebraic Geometry
2024-06-05 v1 Differential Geometry
Abstract
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 K\"ahler metrics in all K\"ahler 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.
Keywords
Cite
@article{arxiv.2011.07135,
title = {The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds},
author = {Thibaut Delcroix},
journal= {arXiv preprint arXiv:2011.07135},
year = {2024}
}