A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves

Álvaro Antón-Sancho; Álvaro Antón-Sancho

doi:10.3934/era.2024283

Electronic Research Archive

2024, Volume 32, Issue 11: 6109-6119. doi: 10.3934/era.2024283

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Research article

A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves

Álvaro Antón-Sancho ^{1,2
,
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1.
Department of Mathematics and Experimental Science, Fray Luis de Leon University College of Education, Catholic University of Ávila, C/Tirso de Molina, 44, 47010 Valladolid, Spain
2.
Technology, Instruction and Design in Engineering and Education Research Group (TiDEE.rg), Catholic University of Ávila, C/Canteros, s/n, 05005 Ávila, Spain

Received: 14 July 2024 Revised: 23 October 2024 Accepted: 06 November 2024 Published: 12 November 2024

Let $ X $ be a compact Riemann surface of genus $ g\geq 2 $ and $ M(G_2) $ be the moduli space of polystable principal $ G_2 $-bundles over $ X $. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space $ M(G_2) $ called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of $ M(G_2) $ corresponding to certain family of Harder-Narasimhan types (specifically, those of the form $ (\lambda, \mu, 0, -\mu, -\lambda) $ with $ \mu < \lambda\leq 0 $) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal $ G_2 $-bundles. From this, it was proved that, when the genus of $ X $ was $ g\geq 12 $, these Shatz strata were the disjoint union of a family of $ G_2 $-Hecke curves in $ M(G_2) $ that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of $ M(G_2) $ through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study.
- principal bundle,
- moduli space,
- $ G_2 $,
- Shatz stratification,
- Hecke curve
Citation: Álvaro Antón-Sancho. A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves[J]. Electronic Research Archive, 2024, 32(11): 6109-6119. doi: 10.3934/era.2024283

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Abstract

Let $ X $ be a compact Riemann surface of genus $ g\geq 2 $ and $ M(G_2) $ be the moduli space of polystable principal $ G_2 $-bundles over $ X $. The Harder-Narasimhan types of the bundles induced a stratification of the moduli space $ M(G_2) $ called Shatz stratification. In this paper, a description of the Shatz strata of the unstable locus of $ M(G_2) $ corresponding to certain family of Harder-Narasimhan types (specifically, those of the form $ (\lambda, \mu, 0, -\mu, -\lambda) $ with $ \mu < \lambda\leq 0 $) was given. For this purpose, a family of vector bundles was constructed in which a 3-form and a 2-form were defined so that it was proved that they were strictly polystable principal $ G_2 $-bundles. From this, it was proved that, when the genus of $ X $ was $ g\geq 12 $, these Shatz strata were the disjoint union of a family of $ G_2 $-Hecke curves in $ M(G_2) $ that will be constructed along the paper. Therefore, the presented results provided an advance in the knowledge of the geometry of $ M(G_2) $ through the study of its Shatz strata and presented a methodological innovation, by using Hecke curves for this study.

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