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On the existence of gr-semistable filtrations of orthogonal/symplectic λ-connections
[Sur l’existence de filtrations gr-semi-stables d’une λ-connexion orthogonale/symplectique]
Mao Sheng1 ; Hao Sun2 ; Jianping Wang3
1 Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China & Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China
2 Department of Mathematics, South China University of Technology, Guangzhou, 510641, China
3 Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China
Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1181-1218.

Dans cet article, nous étudions l’existence de filtrations gr-semi-stables pour des λ-connexions orthogonales/symplectiques. Il est connu que de telles filtrations existent toujours pour les fibrés plats en caractéristique arbitraire. Cependant, nous avons trouvé un contre-exemple pour les fibrés plats orthogonaux de rang 5. La nouvelle idée centrale de cet exemple est la notion de quasi gr-semi-stabilité pour les λ-connexions orthogonales/symplectiques. Nous établissons l’équivalence entre la gr-semi-stabilité et la quasi gr-semi-stabilité pour une λ-connexion orthogonale/symplectique. Cela permet de déterminer si une λ-connexion orthogonale/symplectique est gr-semi-stable. Comme application, nous obtenons une caractérisation des λ-connexions orthogonales gr-semi-stables de rang ≤6.

In this paper, we study the existence of gr-semistable filtrations of orthogonal/symplectic λ-connections. It is known that gr-semistable filtrations always exist for flat bundles in arbitrary characteristic. However, we found a counterexample of orthogonal flat bundles of rank 5. The central new idea in this example is the notion of quasi gr-semistability for orthogonal/symplectic λ-connections. We establish the equivalence between gr-semistability and quasi gr-semistablity for an orthogonal/symplectic λ-connection. This provides a way to determine whether an orthogonal/symplectic λ-connection is gr-semistable. As an application, we obtain a characterization of gr-semistable orthogonal λ-connections of rank ≤6.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.276
Classification : 14D07, 14J60
Keywords: Orthogonal/symplectic $\lambda $-connection, semistability, gr-semistability, quasi gr-semistability
Mot clés : $\lambda $-connexion orthogonale/symplectique, semi-stabilité, gr-semi-stabilité, quasi gr-semi-stabilité

Mao Sheng 1 ; Hao Sun 2 ; Jianping Wang 3

1 Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China & Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China
2 Department of Mathematics, South China University of Technology, Guangzhou, 510641, China
3 Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Mao Sheng; Hao Sun; Jianping Wang. On the existence of gr-semistable filtrations of orthogonal/symplectic $\lambda $-connections. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1181-1218. doi : 10.5802/jep.276. https://jep.centre-mersenne.org/articles/10.5802/jep.276/

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