On the existence of gr-semistable filtrations of orthogonal/symplectic $\lambda $-connections

Sheng, Mao; Sun, Hao; Wang, Jianping

doi:10.5802/jep.276

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]

Sheng, Mao ¹ ; Sun, Hao ² ; Wang, Jianping ³

¹Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China & Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China
²Department of Mathematics, South China University of Technology, Guangzhou, 510641, China
³Yau Mathematical Science Center, Tsinghua University, Beijing, 100084, China

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1181-1218

Abstract (VO)
Résumé

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 $\leq 6$ .

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 $\leq 6$ .

Reçu le : 2024-02-15
Accepté le : 2024-10-09
Publié le : 2024-10-15

MR Zbl

DOI : 10.5802/jep.276

Classification : 14D07, 14J60
Keywords: Orthogonal/symplectic $\lambda $-connection, semistability, gr-semistability, quasi gr-semistability
Mots-clés : $\lambda $-connexion orthogonale/symplectique, semi-stabilité, gr-semi-stabilité, quasi gr-semi-stabilité

Affiliations des auteurs :

Sheng, Mao ¹ ; Sun, Hao ² ; Wang, Jianping 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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     title = {On the existence of gr-semistable filtrations of orthogonal/symplectic $\lambda $-connections},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1181--1218},
     year = {2024},
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Sheng, Mao; Sun, Hao; Wang, Jianping. 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

Bibliographie
Cité par

[AHLH23] Alper, Jarod; Halpern-Leistner, Daniel; Heinloth, Jochen Existence of moduli spaces for algebraic stacks, Invent. Math., Volume 234 (2023) no. 3, pp. 949-1038 | DOI | Zbl | MR

[Ati57] Atiyah, M. F. Vector bundles over an elliptic curve, Proc. London Math. Soc. (3), Volume 7 (1957), pp. 414-452 | DOI | Zbl | MR

[BMW11] Biswas, Indranil; Majumder, Souradeep; Wong, Michael Lennox Orthogonal and symplectic parabolic bundles, J. Geom. Phys., Volume 61 (2011) no. 8, pp. 1462-1475 | DOI | Zbl | MR

[BP03] Balaji, V.; Parameswaran, A. J. Semistable principal bundles. II. Positive characteristics, Transform. Groups, Volume 8 (2003) no. 1, pp. 3-36 | DOI | Zbl | MR

[CH23] Choe, Insong; Hitching, George H. Low rank orthogonal bundles and quadric fibrations, J. Korean Math. Soc., Volume 60 (2023) no. 6, pp. 1137-1169 | DOI | Zbl | MR

[CS21] Collier, Brian; Sanders, Andrew (G,P)-opers and global Slodowy slices, Adv. Math., Volume 377 (2021), 107490, 43 pages | DOI | Zbl | MR

[Gie73] Gieseker, David Stable vector bundles and the Frobenius morphism, Ann. Sci. École Norm. Sup. (4), Volume 6 (1973), pp. 95-101 | Numdam | Zbl | DOI | MR

[GS03] Gómez, Tomas L.; Sols, Ignacio Stable tensors and moduli space of orthogonal sheaves, 2003 | arXiv | arXiv

[HL10] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Cambridge Math. Library, Cambridge University Press, Cambridge, 2010 | DOI | MR

[KSZ21] Kydonakis, Georgios; Sun, Hao; Zhao, Lutian Topological invariants of parabolic $G$ -Higgs bundles, Math. Z., Volume 297 (2021) no. 1-2, pp. 585-632 | DOI | Zbl | MR

[KSZ24] Kydonakis, Georgios; Sun, Hao; Zhao, Lutian Logahoric Higgs torsors for a complex reductive group, Math. Ann., Volume 388 (2024) no. 3, pp. 3183-3228 | DOI | Zbl | MR

[Lan14] Langer, Adrian Semistable modules over Lie algebroids in positive characteristic, Doc. Math., Volume 19 (2014), pp. 509-540 | DOI | Zbl | MR

[Lov17a] Lovering, Tom Integral canonical models for automorphic vector bundles of abelian type, Algebra Number Theory, Volume 11 (2017) no. 8, pp. 1837-1890 | DOI | Zbl | MR

[Lov17b] Lovering, Tom Filtered $F$ -crystals on Shimura varieties of abelian type, 2017 | arXiv

[LSZ19] Lan, Guitang; Sheng, Mao; Zuo, Kang Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups, J. Eur. Math. Soc. (JEMS), Volume 21 (2019) no. 10, pp. 3053-3112 | DOI | Zbl | MR

[Ram75] Ramanathan, A. Stable principal bundles on a compact Riemann surface, Math. Ann., Volume 213 (1975), pp. 129-152 | DOI | Zbl | MR

[Ram96a] Ramanathan, A. Moduli for principal bundles over algebraic curves. I, Proc. Indian Acad. Sci. Math. Sci., Volume 106 (1996) no. 3, pp. 301-328 | DOI | Zbl

[Ram96b] Ramanathan, A. Moduli for principal bundles over algebraic curves. II, Proc. Indian Acad. Sci. Math. Sci., Volume 106 (1996) no. 4, pp. 421-449 | DOI | Zbl | MR

[Sim92] Simpson, Carlos T. Higgs bundles and local systems, Publ. Math. Inst. Hautes Études Sci. (1992) no. 75, pp. 5-95 | Zbl | DOI | Numdam | MR

[Sim97] Simpson, Carlos T. The Hodge filtration on nonabelian cohomology, Algebraic geometry (Santa Cruz, 1995) (Proc. Sympos. Pure Math.), Volume 62, Part 2, American Mathematical Society, Providence, RI, 1997, pp. 217-281 | DOI | Zbl | MR

[Sim10] Simpson, Carlos T. Iterated destabilizing modifications for vector bundles with connection, Vector bundles and complex geometry (Contemp. Math.), Volume 522, American Mathematical Society, Providence, RI, 2010, pp. 183-206 | DOI | Zbl

[Tu93] Tu, Loring W. Semistable bundles over an elliptic curve, Adv. Math., Volume 98 (1993) no. 1, pp. 1-26 | DOI | Zbl | MR

[Yan22] Yang, Mengxue A comparison of generalized opers and $(G, P)$ -opers, Indian J. Pure Appl. Math., Volume 53 (2022) no. 3, pp. 760-773 | DOI | Zbl | MR

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