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The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds

Thibaut Delcroix

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 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.

The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds

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 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.

Paper Structure

This paper contains 16 sections, 8 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Recollections
  3. Cohomogeneity one manifolds and spherical varieties
  4. On rank one spherical varieties
  5. On K-stability of spherical varieties
  6. Uniform K-stability of rank one spherical varieties
  7. Examples
  8. An example of Kähler class with no extremal Kähler metrics
  9. Recollection on the group $G_2$
  10. The facet of the wonderful compactification of $G_2/\mathop{\mathrm{SO}}\nolimits_4$ and its Kähler classes
  11. K-stability condition
  12. Strong Calabi dream manifolds of cohomogeneity one, and an answer to a question of Kanemitsu
  13. Kähler-Einstein metrics on $\mathbb{X}_1$
  14. CscK metrics on the discoloration $\widetilde{\mathbb{X}_1}$
  15. Kähler-Einstein metrics on the $F_4$-variety $\mathbb{X}_2$
  16. ...and 1 more sections

Key Result

Theorem 1.1

On a projective cohomogeneity one manifold, a Kähler class admits a constant scalar curvature Kähler metric if and only if it is K-stable with respect to special equivariant test configurations. The latter amounts to a single combinatorial condition checkable in practice.

Figures (1)

  • Figure 1: Moment polytope $\Delta_+(s,1)$

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm_YTD']}
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • proof : Proof of Theorem \ref{['thm_main']}
  • Theorem 4.1
  • ...and 2 more