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arXiv · 2609.35736

A uniform lower bound for the Zhang-Kawazumi invariant and applications to the Bogomolov conjecture

Robert Wilms
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Abstract

We prove that the Zhang-Kawazumi invariant $φ(X)$ of a compact and connected Riemann surface $X$ of genus $g\ge 2$ is strictly larger than \[\frac{g(g+2)-(2g+1)H_g}{g-1},\] where $H_g=\sum_{k=1}^g \frac{1}{k}$ denotes the $g$-th harmonic number. If $X$ is hyperelliptic, we give the stronger bound $φ(X)>\frac{g}{2}(H_g-1)$. The proof relies on a new expression of $φ(X)$ in terms of a quadratic form on the space of smooth Hermitian matrix-valued functions on $X$, evaluated at certain projector matrices. As an arithmetic application, we deduce new lower bounds for the self-intersection number $ω_a^2$ of the admissible adelic metrized canonical bundle $ω_a$ of a smooth projective curve of genus $g\ge 2$ over a number field and hence new uniform height bounds in the Bogomolov conjecture.

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Robert Wilms. 2026-09-28. A uniform lower bound for the Zhang-Kawazumi invariant and applications to the Bogomolov conjecture. https://arxiv.org/abs/2609.35736

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