[Paper Review] G-uniform stability and K\"{a}hler-Einstein metrics on Fano varieties

Chi Li|arXiv (Cornell University)|Jul 22, 2019

Geometry and complex manifolds67 references28 citations

TL;DR

This paper establishes a G-uniform stability condition for Q-Fano varieties and proves that a Q-Fano variety admits a Kähler-Einstein metric if and only if it is G-uniformly K-stable or Ding-stable, extending the Yau-Tian-Donaldson conjecture to singular Fano varieties with non-discrete automorphism groups. The key innovation is a valuative criterion for G-uniform K-stability using invariant filtrations and non-Archimedean functionals.

ABSTRACT

Let $X$ be any $\\mathbb{Q}$-Fano variety and $\\mathrm{Aut}(X)_0$ be the identity component of the automorphism group of $X$. Let $\\mathbb{G}$ be a connected reductive subgroup of $\\mathrm{Aut}(X)_0$ that contains a maximal torus of $\\mathrm{Aut}(X)_0$. We prove that $X$ admits a K\\"{a}hler-Einstein metric if and only if $X$ is $\\mathbb{G}$-uniformly K-stable. This proves a version of Yau-Tian-Donaldson conjecture for arbitrary singular Fano varieties. A key new ingredient is a valuative criterion for $\\mathbb{G}$-uniform K-stability.

Motivation & Objective

• To extend the Yau-Tian-Donaldson conjecture to Q-Fano varieties with non-discrete automorphism groups.
• To define and analyze G-uniform K-stability for singular Fano varieties, where G is a reductive subgroup of the identity component of the automorphism group.
• To establish a valuative criterion for G-uniform K-stability and Ding-stability using invariant filtrations and non-Archimedean functionals.
• To prove that G-uniform K-stability (or Ding-stability) is equivalent to the existence of a Kähler-Einstein metric on a Q-Fano variety.

Proposed method

• Introduces a G-uniform stability condition for log Fano pairs using test configurations and non-Archimedean functionals, specifically the M^NA and J^NA_T invariants.
• Develops a valuative criterion for G-uniform K-stability by analyzing G-invariant valuations and their associated filtrations on the section ring of the anticanonical line bundle.
• Uses a twisted test configuration construction to perturb destabilizing geodesic rays and establish uniform convergence of L^NA functionals.
• Applies Darvas-Rubinstein's principle on equivariant coercivity to derive an analytic criterion for Kähler-Einstein metrics under group actions.
• Employs a maximal compact subgroup K of G and its Lie algebra decomposition to analyze the structure of the automorphism group and its action on valuations.
• Utilizes the center of the reductive group G and its maximal torus T to define the non-Archimedean J-functional J^NA_T and relate it to the M^NA invariant.

Experimental results

Research questions

• RQ1 Does G-uniform K-stability imply the existence of a Kähler-Einstein metric on a Q-Fano variety with non-discrete automorphism group?
• RQ2 Can a valuative criterion be established for G-uniform K-stability using invariant filtrations and log discrepancies?
• RQ3 Is there a uniform equivalence between G-uniform K-stability, G-uniform Ding-stability, and the existence of a Kähler-Einstein metric?
• RQ4 How do twisted test configurations and perturbed geodesic rays contribute to proving uniform convergence of non-Archimedean functionals?
• RQ5 What is the role of the maximal compact subgroup K and its Lie algebra decomposition in the analytic criterion for Kähler-Einstein metrics?

Key findings

• A Q-Fano variety admits a Kähler-Einstein metric if and only if it is G-uniformly K-stable, where G is a connected reductive subgroup of Aut(X)_0 containing a maximal torus.
• The paper establishes a valuative criterion for G-uniform K-stability, showing that the M^NA invariant is bounded below by a positive multiple of the J^NA_T invariant for all G-equivariant test configurations.
• The equivalence between G-uniform K-stability and G-uniform Ding-stability is proven, extending previous results to singular Fano varieties.
• The existence of a Kähler-Einstein metric is shown to be equivalent to G-uniform Ding-stability, even when the automorphism group is non-discrete.
• The proof relies on constructing a destabilizing geodesic ray and using twisted test configurations to achieve uniform convergence of L^NA functionals.
• The paper confirms the Yau-Tian-Donaldson conjecture in a generalized form for arbitrary singular Fano varieties, not requiring smoothness or discrete automorphism groups.