[Paper Review] Alpha invariant and K-stability of Q-Fano varieties

Yuji Odaka, Yuji Sano|arXiv (Cornell University)|Nov 29, 2010

Geometry and complex manifolds27 references9 citations

TL;DR

This paper establishes a purely algebro-geometric proof that if the log canonical threshold (lct) of a Q-Fano variety X exceeds n/(n+1), then (X, -K_X) is K-stable. The authors link the α-invariant to K-stability via Seshadri constants and log canonical thresholds, proving that high lct implies K-stability and finiteness of the automorphism group, extending Tian's analytic criterion to an algebraic framework.

ABSTRACT

We give a purely algebro-geometric proof that if the alpha-invariant of a Q-Fano variety X is greater than dim X/(dim X+1), then (X,O(-K_X)) is K-stable. The key of our proof is a relation among the Seshadri constants, the alpha-invariant and K-stability. It also gives applications concerning the automorphism group.

Motivation & Objective

• To establish a direct algebro-geometric link between the α-invariant and K-stability of Q-Fano varieties, replacing analytic methods with algebraic invariants.
• To prove that if the log canonical threshold lct(X) > n/(n+1), then (X, O_X(-K_X)) is K-stable, extending Tian's analytic criterion to an algebraic setting.
• To show that K-stability implies finiteness of the automorphism group for smooth Fano manifolds, using Matsushima’s theorem and the absence of nontrivial one-parameter subgroups.
• To generalize the result to G-equivariant K-stability for finite group actions, proving that lct_G(X) > n/(n+1) implies G-equivariant K-stability.

Proposed method

• Define the log canonical threshold lct_G(X) as the infimum over m and G-invariant sublinear systems of |−mK_X|, measuring singularities of pairs (X, 1/m Σ).
• Use Seshadri constants to relate the lct to the geometry of test configurations and the stability of the pair (X, -K_X).
• Construct a G-invariant flag ideal J from a G-equivariant test configuration (X̃, L̃), whose blow-up resolves the rational map X̃ ⇢ X × A¹.
• Apply the log canonical threshold interpretation from [15, Example 9.2.23] to bound the Seshadri constant of the central fiber ideal I₀.
• Use the inequality (n+1)/n K_{B/X×A¹} − Sesh(𝒥, (X×A¹, −K_{X×A¹})) E ≥ 0 to derive a lower bound on the Donaldson-Futaki invariant.
• Leverage Matsushima’s theorem and the fact that K-stability implies no nontrivial one-parameter subgroups in Aut(X) to deduce finiteness of the automorphism group.

Experimental results

Research questions

• RQ1 Does a high log canonical threshold lct(X) > n/(n+1) imply K-stability of a Q-Fano variety X?
• RQ2 Can the analytic criterion of Tian’s α-invariant be fully recovered via algebro-geometric invariants such as lct and Seshadri constants?
• RQ3 What is the relationship between the automorphism group of a Fano manifold and its K-stability, particularly when lct(X) > n/(n+1)?
• RQ4 Is the threshold n/(n+1) sharp for K-stability, or are there exceptions like Pⁿ when equality holds?

Key findings

• If lct(X) > n/(n+1), then (X, O_X(-K_X)) is K-stable for any Q-Fano variety X of dimension n.
• If lct(X) ≥ n/(n+1), then (X, O_X(-K_X)) is K-semistable.
• For smooth Fano manifolds over C, if lct(X) > n/(n+1), then Aut(X) is finite, as K-stability implies no nontrivial one-parameter subgroups.
• The result extends to G-equivariant K-stability: if lct_G(X) > n/(n+1), then (X, O_X(-K_X)) is G-equivariantly K-stable.
• The proof uses a resolution of indeterminacy via blow-up of a G-invariant flag ideal, linking test configurations to Seshadri constants and log canonical thresholds.
• The authors conjecture that equality lct(X) = n/(n+1) implies K-stability unless X ≅ Pⁿ, suggesting the threshold is sharp.