[Paper Review] Remarks on logarithmic K-stability

Chi Li|arXiv (Cornell University)|Apr 3, 2011

Geometry and complex manifolds14 references21 citations

TL;DR

This paper establishes a logarithmic K-stability criterion for toric Fano varieties with a smooth anti-canonical divisor, proving that log-K-stability holds precisely when the cone angle parameter β is less than R(X), the supremum of t for which Ric(ω) > tω is solvable. The key result is a precise characterization of log-K-stability via explicit computation of the log-Futaki invariant using the barycenter and reflexive polytope geometry, confirming a conjecture on conical Kähler-Einstein metrics in the toric setting.

ABSTRACT

We make some observation on the logarithmic version of K-stability.

Motivation & Objective

• To investigate the logarithmic K-stability of toric Fano varieties with a smooth anti-canonical divisor Y.
• To verify Donaldson's conjecture on the existence of Kähler-Einstein metrics with cone singularities along Y for β < R(X).
• To provide a geometric characterization of the threshold β = R(X) in terms of log-Futaki invariants and polytope barycenters.
• To generalize and confirm earlier calculations in [4] by explicitly computing the log-Futaki invariant for 1-parameter subgroups in the torus action.

Proposed method

• Computes R(X) explicitly using Theorem 2, which expresses R(X) as the ratio |OQ|/|P_cQ| where Q lies on the boundary of the reflexive polytope Δ.
• Uses the algebraic definition of the log-Futaki invariant via test configurations associated with 1-parameter subgroups in (C*)^n.
• Derives the main formula (19) for the log-Futaki invariant: F(K_X^{-1}, βY)(λ) = - (β⟨P_c, λ⟩ + (1−β)W(λ)) Vol(Δ).
• Applies convex geometry to analyze the sign of the log-Futaki invariant by comparing Q_β = (β/(1−β))((1−R(X))/R(X)) Q with the supporting hyperplanes of Δ.
• Relies on Wang-Zhu's work and the structure of toric Fano varieties defined by reflexive lattice polytopes.
• Verifies the results on two explicit examples: Bl_pℙ² and Bl_{p,q}ℙ², computing R(X) and log-Futaki invariants for specific 1-parameter subgroups.

Experimental results

Research questions

• RQ1 For which values of β is the pair (X_Δ, βY) log-K-stable along all 1-parameter subgroups in (C*)^n?
• RQ2 What is the precise relationship between the threshold R(X) and the log-Futaki invariant in the logarithmic K-stability condition?
• RQ3 Does the log-Futaki invariant vanish exactly when β = R(X), and if so, for which 1-parameter subgroups?
• RQ4 How does the position of the barycenter P_c and the point Q on ∂Δ determine the stability threshold R(X)?
• RQ5 Can the conjecture on conical Kähler-Einstein metrics be confirmed via log-K-stability in the toric setting?

Key findings

• For β < R(X_Δ), the pair (X_Δ, βY) is log-K-stable along all 1-parameter subgroups in (C*)^n, as the log-Futaki invariant is strictly negative.
• For β = R(X_Δ), the pair is semi-log-K-stable, and the log-Futaki invariant vanishes precisely for 1-parameter subgroups whose supporting hyperplane touches Δ at Q.
• For β > R(X_Δ), the pair is not log-K-stable, as there exists at least one 1-parameter subgroup for which the log-Futaki invariant is positive.
• In the example X_Δ = Bl_pℙ², R(X) = 6/7, and the log-Futaki invariant for λ = ⟨-1,-1⟩ is F = (2/3)β - 4(1−β), which is ≤ 0 iff β ≤ 6/7.
• In the example X_Δ = Bl_{p,q}ℙ², R(X) = 21/25, and the log-Futaki invariant for λ₁ = ⟨1,1⟩ is F = (2/3)β - (7/2)(1−β), which is ≤ 0 iff β ≤ 21/25.
• For λ₃ = ⟨-1,2⟩ in the same example, the invariant allows stability up to β = 63/65, but the critical threshold remains β = 21/25, confirming that β > R(X) implies instability.