# Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups

## TL;DR

This paper provides a counterexample to a conjecture that the Loewy filtration destabilizes Fano varieties with non-reductive automorphism groups, challenging previous assumptions in K-stability theory.

## Contribution

It constructs a specific counterexample to the conjecture relating Loewy filtrations and destabilization of Fano varieties with non-reductive automorphism groups.

## Key findings

- Counterexample disproves the conjecture
- Loewy filtration does not always destabilize such Fano varieties
- Challenges previous beliefs about automorphism groups and K-stability

## Abstract

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan construct a canonical filtration of the section ring, called Loewy filtration, and conjecture that the Loewy filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a counterexample to their conjecture.