A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves
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 descript...
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2024-11-01
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| author | Álvaro Antón-Sancho |
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| description | 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. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2024-11-01 |
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| spelling | doaj-art-a000c7e3d45e438f803fbf3cfc3c03742025-01-23T07:53:00ZengAIMS PressElectronic Research Archive2688-15942024-11-0132116109611910.3934/era.2024283A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curvesÁlvaro Antón-Sancho0Department of Mathematics and Experimental Science, Fray Luis de Leon University College of Education, Catholic University of Ávila, C/Tirso de Molina, 44, 47010 Valladolid, SpainLet $ 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.https://www.aimspress.com/article/doi/10.3934/era.2024283principal bundlemoduli space$ g_2 $shatz stratificationhecke curve |
| spellingShingle | Álvaro Antón-Sancho A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves Electronic Research Archive principal bundle moduli space $ g_2 $ shatz stratification hecke curve |
| title | A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves |
| title_full | A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves |
| title_fullStr | A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves |
| title_full_unstemmed | A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves |
| title_short | A construction of Shatz strata in the polystable $ G_2 $-bundles moduli space using Hecke curves |
| title_sort | construction of shatz strata in the polystable g 2 bundles moduli space using hecke curves |
| topic | principal bundle moduli space $ g_2 $ shatz stratification hecke curve |
| url | https://www.aimspress.com/article/doi/10.3934/era.2024283 |
| work_keys_str_mv | AT alvaroantonsancho aconstructionofshatzstratainthepolystableg2bundlesmodulispaceusingheckecurves AT alvaroantonsancho constructionofshatzstratainthepolystableg2bundlesmodulispaceusingheckecurves |