[Paper Review] Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs

Sébastien Boucksom, Tomoyuki Hisamoto+1 more|arXiv (Cornell University)|Apr 24, 2015

Geometry and complex manifolds87 references196 citations

TL;DR

This paper establishes a non-Archimedean framework for K-stability by introducing non-Archimedean analogues of Kähler functionals, linking Duistermaat-Heckman measures to test configurations, and proving that vanishing $L^p$-norm implies almost triviality. It re-proves and strengthens Y. Odaka’s results on uniform K-stability and singularities of pairs, showing that uniform K-stability is equivalent to the absence of certain singularities in the Minimal Model Program.

ABSTRACT

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in K\\"ahler geometry. These functionals are defined on the space of test configurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and singularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.

Motivation & Objective

• To develop a non-Archimedean formalism for studying K-stability using test configurations and metrics.
• To characterize almost trivial test configurations via Duistermaat-Heckman measures, proving they correspond precisely to Dirac masses.
• To interpret the Donaldson-Futaki invariant as the non-Archimedean analogue of the Mabuchi functional, up to an explicit error term.
• To clarify the relationship between uniform K-stability and singularities of pairs, reproving and strengthening Odaka’s results.

Proposed method

• Define non-Archimedean metrics on the Berkovich analytification of $L$ via test configurations.
• Introduce non-Archimedean analogues of classical Kähler functionals, including the Mabuchi functional.
• Use Duistermaat-Heckman measures to compute the $L^p$-norm of test configurations and characterize their asymptotic weight distributions.
• Apply equivariant Riemann-Roch and asymptotic Riemann-Roch to derive polynomial expansions of weight multiplicities.
• Prove that the Duistermaat-Heckman measure is a Dirac mass if and only if the test configuration is almost trivial.
• Establish that uniform K-stability implies the absence of certain singularities in the Minimal Model Program sense.

Experimental results

Research questions

• RQ1 When is a test configuration almost trivial, and how can this be characterized via its Duistermaat-Heckman measure?
• RQ2 How does the Donaldson-Futaki invariant relate to the non-Archimedean Mabuchi functional?
• RQ3 What is the precise relationship between uniform K-stability and the singularities of pairs in the Minimal Model Program?
• RQ4 Can the $L^p$-norm of a test configuration be computed from its Duistermaat-Heckman measure?
• RQ5 Under what conditions does uniform K-stability imply the absence of non-terminal singularities?

Key findings

• The Duistermaat-Heckman measure of a test configuration is a Dirac mass if and only if the configuration is almost trivial.
• The $L^p$-norm of a test configuration is determined by its Duistermaat-Heckman measure, and vanishing norm implies almost triviality.
• The Donaldson-Futaki invariant equals the non-Archimedean Mabuchi functional up to an explicit error term involving the $L^p$-norm.
• Uniform K-stability is equivalent to the absence of non-terminal singularities in the pair $(X, ext{diff})$ for any log Fano pair.
• The paper provides new examples of uniformly K-stable varieties by analyzing singularities of pairs via the non-Archimedean formalism.
• The non-Archimedean Mabuchi functional is coercive if and only if the pair is uniformly K-stable, generalizing the coercivity conjecture.