Non-reductive automorphism groups, the Loewy filtration and K-stability

TL;DR

This paper investigates the K-stability of polarized varieties with non-reductive automorphism groups by introducing a canonical filtration that often destabilizes the variety, providing insights into the existence of special metrics.

Contribution

It introduces a new canonical filtration of the coordinate ring for non-reductive automorphism groups, linking algebraic stability to geometric properties and conjecturing its general destabilizing effect.

Findings

The filtration destabilizes several examples of such varieties.

Provides an algebro-geometric analogue of Matsushima's theorem.

Constructs an example of an orbifold del Pezzo surface without K"ahler-Einstein metric.

Abstract

We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue of Matsushima's theorem regarding the existence of constant scalar curvature K\"ahler metrics. As an application, we give an example of an orbifold del Pezzo surface without a K\"ahler-Einstein metric.