The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $
Let $ X $ be a compact Riemann surface of genus $ g\geq 2 $. The moduli space $ \mathcal{M}(r, d) $ of rank $ r $ and degree $ d $ semi-stable Higgs bundles over $ X $ admitted a stratification, called Shatz stratification, which was defined by the Harder-Narasimhan type of the Higgs bundles. There...
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2025-02-01
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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 $. The moduli space $ \mathcal{M}(r, d) $ of rank $ r $ and degree $ d $ semi-stable Higgs bundles over $ X $ admitted a stratification, called Shatz stratification, which was defined by the Harder-Narasimhan type of the Higgs bundles. There was also a $ \mathbb{C}^* $-action on $ \mathcal{M}(r, d) $ given by the product on the Higgs field, which provided the Białynicki-Birula stratification by considering the Hodge limit bundles $ \lim_{z\to 0}(E, z\cdot\varphi) $. In this paper, these limit bundles were computed for all possible Harder-Narasimhan types when the rank of the Higgs bundles was $ r = 5 $, explicit vector forms were provided for the Hodge limit bundles, and necessary and sufficient conditions were given for them to be stable. In addition, it was proved that, in rank $ 5 $, the Shatz strata traversed the Białynicki-Birula strata. Specifically, it was checked that there existed different semi-stable rank $ 5 $ Higgs bundles with the same Harder-Narasimhan type such that their associated Hodge limit bundles were not S-equivalent, and explicit constructions of those Higgs bundles were also provided. |
| format | Article |
| id | doaj-art-569d16165b9d4ff9bde0111deea634b1 |
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| language | English |
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| spelling | doaj-art-569d16165b9d4ff9bde0111deea634b12025-08-20T02:26:19ZengAIMS PressAIMS Mathematics2473-69882025-02-011023428345610.3934/math.2025159The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $Álvaro Antón-Sancho0Department of Mathematics and Experimental Science, Fray Luis de León University College of Education, C/Tirso de Molina, 44, 47010 Valladolid, SpainLet $ X $ be a compact Riemann surface of genus $ g\geq 2 $. The moduli space $ \mathcal{M}(r, d) $ of rank $ r $ and degree $ d $ semi-stable Higgs bundles over $ X $ admitted a stratification, called Shatz stratification, which was defined by the Harder-Narasimhan type of the Higgs bundles. There was also a $ \mathbb{C}^* $-action on $ \mathcal{M}(r, d) $ given by the product on the Higgs field, which provided the Białynicki-Birula stratification by considering the Hodge limit bundles $ \lim_{z\to 0}(E, z\cdot\varphi) $. In this paper, these limit bundles were computed for all possible Harder-Narasimhan types when the rank of the Higgs bundles was $ r = 5 $, explicit vector forms were provided for the Hodge limit bundles, and necessary and sufficient conditions were given for them to be stable. In addition, it was proved that, in rank $ 5 $, the Shatz strata traversed the Białynicki-Birula strata. Specifically, it was checked that there existed different semi-stable rank $ 5 $ Higgs bundles with the same Harder-Narasimhan type such that their associated Hodge limit bundles were not S-equivalent, and explicit constructions of those Higgs bundles were also provided.https://www.aimspress.com/article/doi/10.3934/math.2025159higgs bundlestratificationactionharder-narasimhan typesemi-stablemoduli space |
| spellingShingle | Álvaro Antón-Sancho The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $ AIMS Mathematics higgs bundle stratification action harder-narasimhan type semi-stable moduli space |
| title | The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $ |
| title_full | The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $ |
| title_fullStr | The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $ |
| title_full_unstemmed | The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $ |
| title_short | The $ \mathbb{C}^* $-action and stratifications of the moduli space of semi-stable Higgs bundles of rank $ 5 $ |
| title_sort | mathbb c action and stratifications of the moduli space of semi stable higgs bundles of rank 5 |
| topic | higgs bundle stratification action harder-narasimhan type semi-stable moduli space |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025159 |
| work_keys_str_mv | AT alvaroantonsancho themathbbcactionandstratificationsofthemodulispaceofsemistablehiggsbundlesofrank5 AT alvaroantonsancho mathbbcactionandstratificationsofthemodulispaceofsemistablehiggsbundlesofrank5 |