On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910044883255296 |
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| author | Ye, Fei Zhu, Zhixian |
| author_facet | Ye, Fei Zhu, Zhixian |
| contents | In this paper, we study the equivalence between Bogomolov's instability theorem and the Miyaoka-Sakai theorem on surfaces in positive characteristic. We show that Bogomolov's instability theorem can be derived from Miyaoka-Sakai theorem. Conversely, it implies a partial version of the Miyaoka-Sakai theorem that lacks the vanishing conclusion. This partial version is still sufficient to deduce the Mumford-Ramanujam vanishing theorem.
Additionally, we identify a class of surfaces in positive characteristic for which the Miyaoka-Sakai theorem (or a weaker variant), or the Kawamata-Viehweg vanishing theorem holds. In particular, we present a new proof of the Kawamata-Viehweg vanishing theorem on smooth del Pezzo surfaces.
As an application of the Miyaoka-Sakai theorem, we obtain Reider-type results concerning Fujita's conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06975 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic Ye, Fei Zhu, Zhixian Algebraic Geometry In this paper, we study the equivalence between Bogomolov's instability theorem and the Miyaoka-Sakai theorem on surfaces in positive characteristic. We show that Bogomolov's instability theorem can be derived from Miyaoka-Sakai theorem. Conversely, it implies a partial version of the Miyaoka-Sakai theorem that lacks the vanishing conclusion. This partial version is still sufficient to deduce the Mumford-Ramanujam vanishing theorem. Additionally, we identify a class of surfaces in positive characteristic for which the Miyaoka-Sakai theorem (or a weaker variant), or the Kawamata-Viehweg vanishing theorem holds. In particular, we present a new proof of the Kawamata-Viehweg vanishing theorem on smooth del Pezzo surfaces. As an application of the Miyaoka-Sakai theorem, we obtain Reider-type results concerning Fujita's conjecture. |
| title | On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2603.06975 |