Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups
[Exemples de filtrations de Loewy et K-stabilité des variétés de Fano avec groupes d’automorphismes non réductifs]
Annales de l'Institut Fourier, Online first, 23 p.

Il est connu que le groupe d’automorphismes d’une variété de Fano K-polystable est réductif. Codogni et Dervan ont construit une filtration canonique de l’anneau des sections, appelée filtration de Loewy, et ont conjecturé que la filtration déstabilise n’importe quelle variété de Fano avec le groupe d’automorphismes non réductif. Dans cette note, nous fournissons un contre-exemple à leur conjecture.

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

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : https://doi.org/10.5802/aif.3395
Classification : 14J45,  14L30,  32Q26
Mots clés : filtrations de Loewy, K-stabilité
@unpublished{AIF_0__0_0_A46_0,
     author = {Ito, Atsushi},
     title = {Examples on {Loewy} filtrations and {K-stability} of {Fano} varieties with non-reductive automorphism groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3395},
     language = {en},
     note = {Online first},
}
Ito, Atsushi. Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups. Annales de l'Institut Fourier, Online first, 23 p.

[1] Alper, Jarod; Blum, Harold; Halpern-Leistner, Daniel; Xu, Chenyang Reductivity of the automorphism group of K-polystable Fano varieties (2019) (https://arxiv.org/abs/1906.03122, to appear in Invent. Math.) | Article | Zbl 07269010

[2] Berman, Robert J. K-polystability of -Fano varieties admitting Kähler–Einstein metrics, Invent. Math., Volume 203 (2016) no. 3, pp. 973-1025 | Article | MR 3461370 | Zbl 1353.14051

[3] Chen, Xiu Xiong; Donaldson, Simon K.; Sun, Song Kähler–Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 183-197 | Article | MR 3264766 | Zbl 1312.53096

[4] Chen, Xiu Xiong; Donaldson, Simon K.; Sun, Song Kähler–Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 199-234 | Article | MR 3264767 | Zbl 1312.53097

[5] Chen, Xiu Xiong; Donaldson, Simon K.; Sun, Song Kähler–Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof, J. Am. Math. Soc., Volume 28 (2015) no. 1, pp. 235-278 | Article | MR 3264768 | Zbl 1311.53059

[6] Chen, Xiu Xiong; Tian, Gang Geometry of Kähler metrics and foliations by holomorphic discs, Publ. Math., Inst. Hautes Étud. Sci. (2008) no. 107, pp. 1-107 | Article | Numdam | MR 2434691 | Zbl 1182.32009

[7] Codogni, Giulio; Dervan, Ruadhaí Non-reductive automorphism groups, the Loewy filtration and K-stability (2015) (https://arxiv.org/abs/1501.03372v1) | Zbl 1370.32010

[8] Codogni, Giulio; Dervan, Ruadhaí Non-reductive automorphism groups, the Loewy filtration and K-stability, Ann. Inst. Fourier, Volume 66 (2016) no. 5, pp. 1895-1921 | Article | Numdam | MR 3533272 | Zbl 1370.32010

[9] Codogni, Giulio; Dervan, Ruadhaí Corrigendum to “Non-reductive automorphism groups, the Loewy filtration and K-stability”, Ann. Inst. Fourier, Volume 68 (2018) no. 3, pp. 1121-1123 | Article | Numdam | MR 3805769

[10] Cox, David A. The homogeneous coordinate ring of a toric variety, J. Algebr. Geom., Volume 4 (1995) no. 1, pp. 17-50 | MR 1299003 | Zbl 0846.14032

[11] Cox, David A. Erratum to “The homogeneous coordinate ring of a toric variety”, J. Algebr. Geom., Volume 23 (2014) no. 2, pp. 393-398 | Article | MR 3166395 | Zbl 1285.14055

[12] Demazure, Michel Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér., Volume 3 (1970), pp. 507-588 | Article | Numdam | MR 284446 | Zbl 0223.14009

[13] Donaldson, Simon K. Scalar curvature and stability of toric varieties, J. Differ. Geom., Volume 62 (2002) no. 2, pp. 289-349 | MR 1988506 | Zbl 1074.53059

[14] Donaldson, Simon K. Lower bounds on the Calabi functional, J. Differ. Geom., Volume 70 (2005) no. 3, pp. 453-472 | MR 2192937 | Zbl 1149.53042

[15] Matsushima, Yozô Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kählérienne, Nagoya Math. J., Volume 11 (1957), pp. 145-150 | Article | MR 94478 | Zbl 0091.34803

[16] Nill, Benjamin Complete toric varieties with reductive automorphism group, Math. Z., Volume 252 (2006) no. 4, pp. 767-786 | Article | MR 2206625 | Zbl 1091.14011

[17] Sancho, M. T; Moren, J. P; Sancho, Carlos Automorphism group of a toric variety (2018) (https://arxiv.org/abs/1809.09070)

[18] Stoppa, Jacopo K-stability of constant scalar curvature Kähler manifolds, Adv. Math., Volume 221 (2009) no. 4, pp. 1397-1408 | Article | MR 2518643 | Zbl 1181.53060

[19] Tian, Gang Kähler–Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997) no. 1, pp. 1-37 | Article | MR 1471884 | Zbl 0892.53027

[20] Tian, Gang K-stability and Kähler–Einstein metrics, Commun. Pure Appl. Math., Volume 68 (2015) no. 7, pp. 1085-1156 | Article | MR 3352459 | Zbl 1318.14038

[21] Yao, Yi Mabuchi metrics and relative Ding stability of toric Fano varieties (2017) (https://arxiv.org/abs/1701.04016)