Alpha invariants of birationally bi-rigid Fano 3-folds I

TL;DR

This paper calculates the global log canonical thresholds of specific birationally bi-rigid Fano 3-folds in weighted projective spaces, demonstrating their K-stability and existence of orbifold Kähler-Einstein metrics, and provides examples of super-rigid affine Fano 4-folds.

Contribution

It introduces explicit computations of log canonical thresholds for certain Fano 3-folds and establishes their K-stability and metric properties, advancing understanding of their birational geometry.

Findings

Fano 3-folds admit orbifold Kähler-Einstein metrics

These Fano 3-folds are K-stable

Examples of super-rigid affine Fano 4-folds are provided

Abstract

We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold K\"{a}hler-Einstein metric and are K-stable. As an application, we give examples of super-rigid affine Fano 4-folds.