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

Codogni, Giulio; Dervan, Ruadhaí

Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 1895-1921.

Nous étudions la K-stabilité des variétés polarisées telles que leur groupe d’automorphismes ne soit pas réductif. Nous construisons une filtration canonique, que l’on appelle filtration de Loewy, de l’anneau des coordonnées homogènes, qui déstabilise la variété dans beaucoup d’exemples. Nous conjecturons que cette filtration déstabilise toutes les variétés avec groupe d’automorphismes non réductif. Ceci est un analogue algébro-géométrique du théorème de Matsushima sur la non-existence de métriques Kahleriennes avec courbure scalaire constante sur les variétés avec un groupe d’automorphismes non réductif. En tant qu’application, nous donnons un exemple de surface orbifolde de del Pezzo qui n’admet pas de métrique de Kähler-Einstein.

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 Kähler metrics. As an application, we give an example of an orbifold del Pezzo surface without a Kähler-Einstein metric.

Classification : 32Q26, 32M99, 32Q20, 17B20
Keywords: K-stability, reductive groups, Kähler-Einstein metrics, radical filtration
Mot clés : K-stabilité, groupes reductif, métriques de Kähler-Einstein, filtration radical

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