# Reductivity of the automorphism group of K-polystable Fano varieties

## TL;DR

This paper proves that K-polystable log Fano pairs have reductive automorphism groups and explores the moduli space structure of K-semistable Fano varieties, advancing understanding of their geometric properties.

## Contribution

It establishes reductivity of automorphism groups for K-polystable log Fano pairs and analyzes the moduli space structure under K-semistability assumptions.

## Key findings

- Automorphism groups of K-polystable log Fano pairs are reductive.
- The moduli stack of K-semistable Fano varieties admits a separated good moduli space.
- Results depend on the conjecture that K-semistability is an open condition.

## Abstract

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.