A Matsushima theorem for K-polystable polarised smooth Fano threefolds

TL;DR

This paper proves that for smooth Fano threefolds with a K-polystable ample divisor, the automorphism group is reductive, confirming a key aspect of the Yau--Tian--Donaldson conjecture.

Contribution

It establishes the reductivity of automorphism groups for K-polystable smooth Fano threefolds with arbitrary polarizations, confirming a predicted conjecture.

Findings

Automorphism group of K-polystable smooth Fano threefolds is reductive.

Verifies the Yau--Tian--Donaldson conjecture in this setting.

Supports the link between K-stability and geometric symmetry properties.

Abstract

We prove that if $X$ is a smooth Fano threefold and $L$ is an ample $\mathbb{Q}$-divisor such that $(X,L)$ is K-polystable, then the automorphism group $\operatorname{Aut}(X)$ is reductive. This verifies the reductivity statement predicted by the Yau--Tian--Donaldson conjecture in the setting of smooth Fano threefolds with arbitrary ample polarisation.