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Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka

Yoshinori Hashimoto

Abstract

We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein-Tian-Zhang, we obtain the following algebro-geometric corollary: the $δ_m$-invariant of Fujita-Odaka satisfies $δ_m >1$ if and only if the Fano manifold is stable in the sense of Saito-Takahashi, establishing a Hilbert-Mumford type criterion for $δ_m >1$. We also extend this result to the Kähler-Ricci $g$-solitons and the coupled Kähler-Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.

Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka

Abstract

We prove that the stability condition for Fano manifolds defined by Saito-Takahashi, given in terms of the sum of the Ding invariant and the Chow weight, is equivalent to the existence of anticanonically balanced metrics. Combined with the result by Rubinstein-Tian-Zhang, we obtain the following algebro-geometric corollary: the -invariant of Fujita-Odaka satisfies if and only if the Fano manifold is stable in the sense of Saito-Takahashi, establishing a Hilbert-Mumford type criterion for . We also extend this result to the Kähler-Ricci -solitons and the coupled Kähler-Einstein metrics, and as a by-product we obtain a formula for the asymptotic slope of the coupled Ding functional in terms of multiple test configurations.

Paper Structure

This paper contains 23 sections, 21 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of the results
  4. Organisation of the paper
  5. Differential-geometric preliminaries
  6. Kähler--Einstein metrics and Ding functional
  7. Anticanonically balanced metrics
  8. Kähler--Ricci $g$-solitons and balanced metrics
  9. Coupled Kähler--Einstein metrics and balanced metrics
  10. Algebro-geometric preliminaries
  11. Test configurations
  12. Ding invariant and Chow weight
  13. Kähler--Ricci $g$-solitons case
  14. Coupled Ding invariant
  15. Psh rays, convexity, and the slope formulae
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $m \in \mathbb{N}$ be large enough such that $-mK_X$ is very ample. A Fano manifold $(X , -K_X)$ admits an anticanonically balanced metric at level $m$, which is unique up to $\mathrm{Aut}_0(X)$, if and only if it satisfies the following stability condition: for any very ample test configuration

Theorems & Definitions (77)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 67 more