no. 2
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]
Ito, Atsushi1
1 Nagoya University Graduate School of Mathematics Furocho Chikusaku Nagoya, 464-8602 (Japan)
Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 515-537.

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 :
Publié le :
DOI : 10.5802/aif.3395
Classification : 14J45, 14L30, 32Q26
Keywords: Loewy filtration, K-stability
Mot clés : filtrations de Loewy, K-stabilité

Ito, Atsushi 1

1 Nagoya University Graduate School of Mathematics Furocho Chikusaku Nagoya, 464-8602 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2021__71_2_515_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},
     pages = {515--537},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {2},
     year = {2021},
     doi = {10.5802/aif.3395},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3395/}
}
TY  - JOUR
AU  - Ito, Atsushi
TI  - Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 515
EP  - 537
VL  - 71
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3395/
DO  - 10.5802/aif.3395
LA  - en
ID  - AIF_2021__71_2_515_0
ER  - 
%0 Journal Article
%A Ito, Atsushi
%T Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups
%J Annales de l'Institut Fourier
%D 2021
%P 515-537
%V 71
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3395/
%R 10.5802/aif.3395
%G en
%F AIF_2021__71_2_515_0
Ito, Atsushi. Examples on Loewy filtrations and K-stability of Fano varieties with non-reductive automorphism groups. Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 515-537. doi : 10.5802/aif.3395. https://aif.centre-mersenne.org/articles/10.5802/aif.3395/

[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.) | DOI

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

[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 | DOI | Numdam | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

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

Cité par Sources :