[Paper Review] Geodesic rays and stability in the cscK problem

Chi Li|arXiv (Cornell University)|Jan 6, 2020

Geometry and complex manifolds73 references26 citations

TL;DR

This paper establishes that any finite energy geodesic ray with finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, reducing the uniform Yau-Tian-Donaldson conjecture to Boucksom-Jonsson's conjecture on algebraic approximation of non-Archimedean entropy. It further shows that uniform K-stability for model filtrations and $\mathcal{J}^{K_X}$-stability are sufficient conditions for the existence of cscK metrics, with applications to toric manifolds and new identities for Mabuchi slopes.

ABSTRACT

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archimedean entropy functional. As further applications, we show that a uniform K-stability condition for model filtrations and the $\mathcal{J}^{K_X}$-stability are both sufficient conditions for the existence of cscK metrics. The first condition is also conjectured to be necessary. Our arguments also produce a different proof of the toric uniform version of YTD conjecture for all polarized toric manifolds. Another result proved here is that the Mabuchi slope of a geodesic ray associated to a test configuration is equal to the non-Archimedean Mabuchi invariant.

Motivation & Objective

• To establish a link between destabilizing geodesic rays and algebraic approximability via maximality.
• To reduce the uniform Yau-Tian-Donaldson conjecture to the conjecture on algebraic approximation of non-Archimedean entropy.
• To prove that uniform $\mathcal{J}^{K_X}$-stability and uniform stability for model filtrations are sufficient conditions for the existence of cscK metrics.
• To verify the Mabuchi slope identity for geodesic rays associated to test configurations.
• To provide a new proof of the toric uniform YTD conjecture for all polarized toric manifolds.

Proposed method

• Uses the convexity of the Mabuchi energy along $C^{1,\bar{1}}$-geodesics to ensure existence of the Mabuchi slope limit.
• Applies the Berman-Boucksom-Jonsson correspondence between maximal geodesic rays and finite energy non-Archimedean metrics.
• Employs strong convergence of non-Archimedean metrics $\phi_m \to \Phi_{\rm NA}$ to analyze limits of twisted Monge-Ampère energy slopes.
• Relies on estimates from [8], [3], and [5] to prove convergence of twisted Monge-Ampère energy slopes for maximal geodesic rays.
• Uses Chen-Tian decomposition of Mabuchi energy into entropy and energy parts to isolate the entropy slope problem.
• Applies known results on properness of $\mathcal{J}^{K_X}$-energy and solvability of the $J$-equation to derive existence theorems.

Experimental results

Research questions

• RQ1 Is every finite energy geodesic ray with finite Mabuchi slope maximal in the sense of Berman-Boucksom-Jonsson?
• RQ2 Can the Mabuchi slope of a maximal geodesic ray be algebraically approximated via test configurations?
• RQ3 Does uniform $\mathcal{J}^{K_X}$-stability imply the existence of a cscK metric?
• RQ4 Is the Mabuchi slope of a geodesic ray associated to a test configuration equal to the non-Archimedean Mabuchi invariant?
• RQ5 Does the convergence of twisted Monge-Ampère energy slopes for maximal geodesic rays hold under strong convergence of non-Archimedean metrics?

Key findings

• Any finite energy geodesic ray with finite Mabuchi slope is maximal, and satisfies $\mathbf{E}^{\prime\infty}(\Phi) = \mathbf{E}^{\rm NA}(\Phi_{\rm NA})$.
• The Mabuchi slope of a geodesic ray associated to a test configuration equals the non-Archimedean Mabuchi invariant.
• The convergence $({\bf E}^{{\rm dd^{c}}\psi_{Q}})^{\prime\infty}(\Phi) = \lim_{m\to\infty}({\bf E}^{{\rm dd^{c}}\psi_{Q}})^{\prime\infty}(\Phi_m)$ holds for maximal geodesic rays.
• Uniform $\mathcal{J}^{K_X}$-stability implies the existence of a cscK metric.
• Uniform stability for model filtrations is also a sufficient condition for the existence of cscK metrics.
• The identity $({\bf E}^{{\rm dd^{c}}\psi_{Q}})^{\prime\infty}(\Phi) = ({\bf E}^{Q_{\mathbb{C}}})^{\rm NA}(\Phi_{\rm NA})$ holds for any maximal geodesic ray.