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A YTD correspondence for constant scalar curvature metrics

Tamás Darvas, Kewei Zhang

Abstract

Given a compact Kähler manifold, to better understand Mabuchi's $K$ energy we introduce a family of $K^β$ energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the $K^β$ energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau-Tian-Donaldson correspondence that characterizes the existence of a unique constant scalar curvature Kähler metric using test configurations. Combining our techniques with the non-Archimedean approach to $K$-stability pioneered by Boucksom--Jonsson, we show that the properness of the classical $K$ energy can be tested by checking its slope along a distinguished subclass of Chi Li-type models, called log discrepancy models, thus yielding another $G$-uniform Yau--Tian--Donaldson correspondence.

A YTD correspondence for constant scalar curvature metrics

Abstract

Given a compact Kähler manifold, to better understand Mabuchi's energy we introduce a family of energies, whose favorable properties are similar to those of the Ding energy from the Fano case. The construction uses Berman's transcendental quantization, and we show that the slope of the energies along test configurations can be computed using intersection theory. With these ingredients in place we provide a uniform Yau-Tian-Donaldson correspondence that characterizes the existence of a unique constant scalar curvature Kähler metric using test configurations. Combining our techniques with the non-Archimedean approach to -stability pioneered by Boucksom--Jonsson, we show that the properness of the classical energy can be tested by checking its slope along a distinguished subclass of Chi Li-type models, called log discrepancy models, thus yielding another -uniform Yau--Tian--Donaldson correspondence.

Paper Structure

This paper contains 28 sections, 52 theorems, 241 equations.

Table of Contents

  1. The main results
  2. The $K^\beta$ energy and transcendental quantization.
  3. A YTD correspondence using $K^\beta$.
  4. Algebraic interpretation of $K^\beta_\mathcal{T}$.
  5. Interactions with the non-Archimedean point of view.
  6. Acknowledgments.
  7. Preliminaries
  8. Pluripotential theory.
  9. Test configurations and geodesic rays.
  10. Compactification and domination.
  11. Demailly regularization of rays.
  12. Models and non-Archimedean geometry.
  13. Complex reductive Lie groups.
  14. Transcendental quantization
  15. The $\beta$-entropy and quantization.
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $(X,L)$ be a polarized Kähler manifold. The following are equivalent. (i) $(X,L)$ admits a unique cscK metric in $c_1(L)$. (ii) $(X,L)$ is uniformly $K^\beta$-stable for some $\beta>0$.

Theorems & Definitions (99)

  • Theorem 1.1
  • proof : Proof of Theorem \ref{["thm:YTD'"]}
  • Theorem 1.2: Theorem \ref{["thm:K_beta_intersection'"]}
  • Theorem 1.3
  • Conjecture 1.4: Boucksom--Jonsson
  • Theorem 1.5: Theorem \ref{['thm:BJ-conj_main1']}
  • Theorem 1.6: Theorem \ref{['thm:YTD"\'1']}
  • Remark 1.7
  • Lemma 2.1
  • Lemma 3.1
  • ...and 89 more