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

Abstract

We prove that the Zhang-Kawazumi invariant $\varphi(X)$ 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 $\varphi(X)>\frac{g}{2}(H_g-1)$. The proof relies on a new expression of $\varphi(X)$ 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 $\omega_a^2$ of the admissible adelic metrized canonical bundle $\omega_a$ of a smooth projective curve of genus $g\ge 2$ over a number field and hence new uniform height bounds in the Bogomolov conjecture.