[Paper Review] Alpha invariants of birationally rigid Fano threefolds

In‐Kyun Kim, Takuzo Okada+1 more|arXiv (Cornell University)|Apr 1, 2016

Geometry and complex manifolds16 references3 citations

TL;DR

This paper computes the global log canonical threshold (alpha invariant) for birationally rigid orbifold Fano threefolds embedded in weighted projective spaces of codimension two and three, proving that all such threefolds except one family have alpha invariant exactly 1. This implies K-stability and the existence of a Kähler–Einstein metric, resolving key stability and geometric structure questions for these Fano threefolds.

ABSTRACT

We compute global log canonical thresholds, or equivalently alpha invariants, of birationally rigid orbifold Fano threefolds embedded in weighted projective spaces as codimension two or three. As an important application, we prove that most of them are weakly exceptional, $K$-stable and admit Kähler--Einstien metric.

Motivation & Objective

• To compute the global log canonical threshold (alpha invariant) for birationally rigid Fano threefolds of codimension two and three in weighted projective spaces.
• To establish K-stability and the existence of Kähler–Einstein metrics for these Fano threefolds using alpha invariant bounds.
• To determine whether these Fano threefolds are weakly exceptional by analyzing their singularities and birational geometry.
• To complete the classification of alpha invariants for all known families of quasi-smooth Fano threefolds of index one in codimension two and three.
• To extend the understanding of birational rigidity and singularities in Fano threefolds via log canonical threshold techniques.

Proposed method

• The authors compute the global log canonical threshold lct(X) by analyzing the log canonical threshold at singular points and using adjunction theory on the Fano threefold X.
• They apply Lemma 2.7 and Lemma 2.5 to bound multiplicities and orders of divisors, particularly focusing on the behavior of hyperplane sections and their intersections.
• The index cover construction is used to resolve singularities of type 1/5(1,2,3), enabling analysis of the multiplicity of curves at singular points.
• The method relies on the structure of the anticanonical divisor and the use of numerical equivalence to control the behavior of Q-divisors numerically equivalent to -K_X.
• The proof uses the fact that lct(X) > dim X / (dim X + 1) implies K-stability and existence of a Kähler–Einstein metric, linking analytic and algebro-geometric notions.
• For codimension 3 Fano threefolds of degree 1/20, the authors analyze the index 1 cover and use coordinate-specific equations to compute the multiplicity of the curve L_xy at the singular point.

Experimental results

Research questions

• RQ1 What is the global log canonical threshold of a birationally rigid Fano threefold of codimension two or three in a weighted projective space?
• RQ2 Does the alpha invariant of such Fano threefolds exceed the critical threshold dim X / (dim X + 1), implying K-stability and existence of a Kähler–Einstein metric?
• RQ3 Are these Fano threefolds weakly exceptional, meaning they admit a unique plt blow-up at their singularities?
• RQ4 How do the multiplicities of hyperplane sections and their intersections behave at singular points of type 1/5(1,2,3)?
• RQ5 Can the index cover method be effectively used to compute log canonical thresholds for Fano threefolds with non-terminal quotient singularities?

Key findings

• For all birationally rigid Fano threefolds of codimension two (except the complete intersection of quadric and cubic in P^5), the global log canonical threshold satisfies lct(X) ≥ 1, with equality holding except for one specific family (No. 60).
• For all birationally rigid Fano threefolds of codimension three, the global log canonical threshold is exactly 1, i.e., lct(X) = 1.
• The condition lct(X) = 1 implies that these Fano threefolds are K-stable and admit a Kähler–Einstein metric.
• The singular point of type 1/5(1,2,3) on codimension three Fano threefolds of degree 1/20 has multiplicity 2 in the index cover, which is crucial for threshold computation.
• The variety V = X_1 × ⋯ × X_r for such Fano threefolds is non-rational and its birational automorphism group is generated by the product of individual birational automorphism groups and automorphisms of V.
• The singularity at the origin of the affine cone over these Fano threefolds is weakly exceptional, as confirmed by the unique plt blow-up property.