[Paper Review] Testing log K-stability by blowing up formalism

Yuji Odaka, Song Sun|arXiv (Cornell University)|Dec 6, 2011

Geometry and complex manifolds17 references13 citations

TL;DR

This paper extends the theory of K-stability to the logarithmic setting by generalizing the Donaldson-Futaki invariant formula for pairs $(X,D)$ with cone angle $2etaeta o 0$. It establishes that log K-stability implies semi-log-canonicity of $(X,(1-eta)D)$, and proves that log K-stability is equivalent to semi-log-canonicity for log Calabi-Yau and log canonical pairs, providing algebraic counterparts to results on Kähler-Einstein metrics with cone singularities.

ABSTRACT

We study logarithmic K-stability for pairs by extending the formula for Donaldson-Futaki invariants to log setting. We also provide algebro-geometric counterparts of recent results of existence of Kahler-Einstein metrics with cone singularities.

Motivation & Objective

• To extend the theory of K-stability to logarithmic pairs $(X,D)$ with cone angle $2\pi\beta$.
• To generalize the Donaldson-Futaki invariant formula to the log setting for pairs with a boundary divisor $D$.
• To establish algebraic counterparts of recent results on Kähler-Einstein metrics with cone singularities.
• To prove that log K-semistability implies semi-log-canonicity of $(X,(1-\beta)D)$, and to characterize log K-stability in terms of singularities.
• To unify the minimal model program (MMP) framework with log K-stability via blow-up formalism and test configurations.

Proposed method

• Adapts the Donaldson-Futaki invariant to log pairs by introducing a modified formula involving discrepancies and valuations of ideals.
• Uses log resolutions and the negativity lemma to analyze the behavior of canonical divisors under blow-ups of ideals.
• Applies the $S_2$ condition and integral closure of ideals to ensure compatibility with the definition of K-stability.
• Constructs flag ideals $\mathcal{J}$ on the product $X \times \mathbb{A}^1$ to model test configurations and compute log DF invariants.
• Employs the semi-log-canonical model $\pi: B \to (X, (1-\beta)D)$ to analyze singularities and derive instability when the pair is not semi-log-canonical.
• Uses the leading coefficient of the log DF invariant in $r$ to detect instability, particularly when $\mathcal{L}^{n-1} \cdot E^2 < 0$.

Experimental results

Research questions

• RQ1 How can the Donaldson-Futaki invariant be generalized to the logarithmic setting for pairs $(X,D)$ with cone angle $2\pi\beta$?
• RQ2 What is the precise relationship between log K-stability and the singularities of the pair $(X,(1-\beta)D)$?
• RQ3 Can algebraic counterparts of results on Kähler-Einstein metrics with cone singularities be established via log K-stability?
• RQ4 Under what conditions does log K-semistability imply semi-log-canonicity of $(X,(1-\beta)D)$?
• RQ5 How does the minimal model program (MMP) framework interact with log K-stability through blow-up formalism?

Key findings

• The log Donaldson-Futaki invariant is extended to pairs $(X,D)$ via a formula involving discrepancies and ideal valuations, enabling stability testing in the log setting.
• Log K-semistability with cone angle $2\pi\beta$ implies that $(X,(1-\beta)D)$ is semi-log-canonical, generalizing results from the absolute case.
• For log $\mathbb{Q}$-Fano varieties, log K-semistability implies that $(X,(1-\beta)D)$ is kawamata log terminal (klt) when $\beta > 0$.
• In the log Calabi-Yau case, $((X,D),L)$ is log K-semistable if and only if $(X,(1-\beta)D)$ is semi-log-canonical.
• For log canonical pairs, log K-stability, log K-semistability, and semi-log-canonicity of $(X,(1-\beta)D)$ are equivalent.
• The paper provides algebraic proofs of results previously obtained via analytic methods, such as those in [Ber10], [Bre11], and [JMR11], strengthening the Yau-Tian-Donaldson conjecture in the log setting.