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arXiv · 2512.01811

Numerical cohomology for arithmetic surfaces and applications

Wei He
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Abstract

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.

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Wei He. 2025-12-01. Numerical cohomology for arithmetic surfaces and applications. https://arxiv.org/abs/2512.01811

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