Numerical cohomology for arithmetic surfaces and applications
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912742172000256 |
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| author | He, Wei |
| author_facet | He, Wei |
| contents | In this paper, we introduce numerical cohomology for arithmetic surfaces, which leads to an absolute version of arithmetic Riemann-Roch formula. As an application, we derive an upper bound for the self-intersection number of relative dualizing sheaf in terms of successive minima with respect to $L^2$-norm. The result has the geometric analogue that the slopes of the Harder-Narasimhan filtration of relative dualizing sheaf provide an upper bound for self-intersection number. Suppose that the arithmetic surface admits a section and has generic fiber of genus at least two, we obtain a refined upper bound for the self-intersection number, which is governed by the topological and arithmetic information of the section. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01811 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numerical cohomology for arithmetic surfaces and applications He, Wei Number Theory Algebraic Geometry In this paper, we introduce numerical cohomology for arithmetic surfaces, which leads to an absolute version of arithmetic Riemann-Roch formula. As an application, we derive an upper bound for the self-intersection number of relative dualizing sheaf in terms of successive minima with respect to $L^2$-norm. The result has the geometric analogue that the slopes of the Harder-Narasimhan filtration of relative dualizing sheaf provide an upper bound for self-intersection number. Suppose that the arithmetic surface admits a section and has generic fiber of genus at least two, we obtain a refined upper bound for the self-intersection number, which is governed by the topological and arithmetic information of the section. |
| title | Numerical cohomology for arithmetic surfaces and applications |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2512.01811 |