Frobenius pull-back of parabolic bundles and dormant opers
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908519258652672 |
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| author | Wakabayashi, Yasuhiro |
| author_facet | Wakabayashi, Yasuhiro |
| contents | We study parabolic bundles on an algebraic curve in positive characteristic. Our motivation is to properly formulate Frobenius pull-backs of parabolic bundles in a way that extends various previous facts and arguments for the usual non-parabolic Frobenius pull-backs. After defining that operation, we generalize a classical result by Cartier concerning Frobenius descent, that is, we establish a bijective correspondence (including the version using higher-level $\mathcal{D}$-modules) between parabolic flat bundles with vanishing $p$-curvature on a pointed curve and parabolic bundles on its Frobenius twist. This correspondence gives a description of maximally Frobenius-destabilized parabolic bundles in terms of dormant opers admitting logarithmic poles. As an application of that description together with a previous result in the enumerative geometry of dormant opers, we obtain an explicit formula for computing the number of such parabolic bundles of rank $2$ under certain assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12267 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Frobenius pull-back of parabolic bundles and dormant opers Wakabayashi, Yasuhiro Algebraic Geometry We study parabolic bundles on an algebraic curve in positive characteristic. Our motivation is to properly formulate Frobenius pull-backs of parabolic bundles in a way that extends various previous facts and arguments for the usual non-parabolic Frobenius pull-backs. After defining that operation, we generalize a classical result by Cartier concerning Frobenius descent, that is, we establish a bijective correspondence (including the version using higher-level $\mathcal{D}$-modules) between parabolic flat bundles with vanishing $p$-curvature on a pointed curve and parabolic bundles on its Frobenius twist. This correspondence gives a description of maximally Frobenius-destabilized parabolic bundles in terms of dormant opers admitting logarithmic poles. As an application of that description together with a previous result in the enumerative geometry of dormant opers, we obtain an explicit formula for computing the number of such parabolic bundles of rank $2$ under certain assumptions. |
| title | Frobenius pull-back of parabolic bundles and dormant opers |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2408.12267 |