K-stability of Fano 3-folds in the World of Null-A

TL;DR

This paper investigates the K-stability of smooth Fano 3-folds, establishing conditions under which they are K-polystable based on automorphism group properties and classifying exceptional cases.

Contribution

It introduces a criterion involving Condition (A) for K-polystability of Fano 3-folds and classifies the exceptional deformation families where this does not hold.

Findings

Fano 3-folds not satisfying Condition (A) are generally K-polystable.

Eight exceptional deformation families are identified where K-polystability may not hold.

Most smooth Fano 3-folds are K-polystable outside these exceptions.

Abstract

A variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We show that a smooth Fano 3-fold not satisfying Condition (A) is K-polystable unless it is contained in eight exceptional deformation families (seven of them consists of one smooth member, and one of them has one-parameter moduli).