Non-reductive automorphism groups, the Loewy filtration and K-stability
Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 1895-1921.

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.

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.

Revised after acceptance:
Published online:
DOI: 10.5802/aif.3052
Classification: 32Q26, 32M99, 32Q20, 17B20
Keywords: K-stability, reductive groups, Kähler-Einstein metrics, radical filtration
Codogni, Giulio 1; Dervan, Ruadhaí 2

1 Dipartimento di Matematica e Fisica Università Roma Tre Largo San Leonardo Murialdo 1 00146 Roma (Italy)
2 University of Cambridge DPMMS Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB (United Kingdom)
     author = {Codogni, Giulio and Dervan, Ruadha{\'\i}},
     title = {Non-reductive automorphism groups, the {Loewy} filtration and {K-stability}},
     journal = {Annales de l'Institut Fourier},
     pages = {1895--1921},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {5},
     year = {2016},
     doi = {10.5802/aif.3052},
     language = {en},
     url = {}
AU  - Codogni, Giulio
AU  - Dervan, Ruadhaí
TI  - Non-reductive automorphism groups, the Loewy filtration and K-stability
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 1895
EP  - 1921
VL  - 66
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  -
UR  -
DO  - 10.5802/aif.3052
LA  - en
ID  - AIF_2016__66_5_1895_0
ER  - 
%0 Journal Article
%A Codogni, Giulio
%A Dervan, Ruadhaí
%T Non-reductive automorphism groups, the Loewy filtration and K-stability
%J Annales de l'Institut Fourier
%D 2016
%P 1895-1921
%V 66
%N 5
%I Association des Annales de l’institut Fourier
%R 10.5802/aif.3052
%G en
%F AIF_2016__66_5_1895_0
Codogni, Giulio; Dervan, Ruadhaí. Non-reductive automorphism groups, the Loewy filtration and K-stability. Annales de l'Institut Fourier, Volume 66 (2016) no. 5, pp. 1895-1921. doi : 10.5802/aif.3052.

[1] Assem, Ibrahim; Simson, Daniel; Skowroński, Andrzej Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006, x+458 pages (Techniques of representation theory)

[2] Berman, Robert J. K-polystability of Q-Fano varieties admitting Kahler-Einstein metrics (

[3] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI

[4] Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (

[5] Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics and stability, Int. Math. Res. Not. IMRN (2014) no. 8, pp. 2119-2125

[6] Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof, J. Amer. Math. Soc., Volume 28 (2015) no. 1, pp. 235-278 | DOI

[7] Flips for 3-folds and 4-folds (Corti, Alessio, ed.), Oxford Lecture Series in Mathematics and its Applications, 35, Oxford University Press, Oxford, 2007, x+189 pages | DOI

[8] Debarre, Olivier Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001, xiv+233 pages

[9] Dervan, R. Uniform stability of twisted constant scalar curvature Kähler metrics (2015) (to appear in Int. Math. Res. Not. IMRN)

[10] Donaldson, S. K. Scalar curvature and projective embeddings. I, J. Differential Geom., Volume 59 (2001) no. 3, pp. 479-522

[11] Donaldson, S. K. Scalar curvature and stability of toric varieties, J. Differential Geom., Volume 62 (2002) no. 2, pp. 289-349

[12] Donaldson, S. K. Lower bounds on the Calabi functional, J. Differential Geom., Volume 70 (2005) no. 3, pp. 453-472

[13] Donaldson, S. K. Stability, birational transformations and the Kahler-Einstein problem, Surveys in differential geometry. Vol. XVII (Surv. Differ. Geom.), Volume 17, Int. Press, Boston, MA, 2012, pp. 203-228 | DOI

[14] Gross, Daniel On the fundamental group of the fixed points of a unipotent action, Manuscripta Math., Volume 60 (1988) no. 4, pp. 487-496 | DOI

[15] Horrocks, G. Fixed point schemes of additive group actions, Topology, Volume 8 (1969), pp. 233-242 | DOI

[16] Jantzen, Jens Carsten Representations of algebraic groups, Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003, xiv+576 pages

[17] Jelonek, Z.; Lasoń, M. The set of fixed points of a unipotent group, J. Algebra, Volume 322 (2009) no. 6, pp. 2180-2185 | DOI

[18] Levine, Marc A remark on extremal Kähler metrics, J. Differential Geom., Volume 21 (1985) no. 1, pp. 73-77

[19] Li, Chi; Xu, Chenyang Special test configuration and K-stability of Fano varieties, Ann. of Math. (2), Volume 180 (2014) no. 1, pp. 197-232 | DOI

[20] Matsushima, Yozô Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math. J., Volume 11 (1957), pp. 145-150 | DOI

[21] Panov, Dmitri; Ross, Julius Slope stability and exceptional divisors of high genus, Math. Ann., Volume 343 (2009) no. 1, pp. 79-101 | DOI

[22] Ross, Julius; Thomas, Richard A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom., Volume 16 (2007) no. 2, pp. 201-255 | DOI

[23] Stoppa, Jacopo A note on the definition of K-stability (

[24] Stoppa, Jacopo K-stability of constant scalar curvature Kähler manifolds, Adv. Math., Volume 221 (2009) no. 4, pp. 1397-1408 | DOI

[25] Székelyhidi, Gábor Filtrations and test-configurations (2011) (

[26] Székelyhidi, Gábor Extremal Kähler metrics (2014) (Proceedings of the ICM,

[27] Tian, Gang K-stability and Kähler-Einstein metrics, Comm. Pure Appl. Math., Volume 68 (2015) no. 7, pp. 1085-1156 | DOI

[28] Witt Nyström, David Test configurations and Okounkov bodies, Compos. Math., Volume 148 (2012) no. 6, pp. 1736-1756 | DOI

Cited by Sources: