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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≥ 2 is strictly larger than (g(g+2)−(2g+1)Hg)/(g−1), where Hg=∑k=1g 1/k denotes the g-th harmonic number. If X is hyperelliptic, we give the stronger bound φ(X)>g/2(Hg−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 ωa2 of the admissible adelic metrized canonical bundle ωa of a smooth projective curve of genus g≥ 2 over a number field and hence new uniform height bounds in the Bogomolov conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{wilms2026uniform,
  title = {{A uniform lower bound for the Zhang-Kawazumi invariant and applications to the Bogomolov conjecture}},
  author = {Robert Wilms},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35736},
  eprint = {2609.35736},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35736},
}
RIS
TY  - UNPB
TI  - A uniform lower bound for the Zhang-Kawazumi invariant and applications to the Bogomolov conjecture
AU  - Robert Wilms
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35736
UR  - https://arxiv.org/abs/2609.35736
AB  - We prove that the Zhang-Kawazumi invariant φ(X) of a compact and connected Riemann surface X of genus g≥ 2 is strictly larger than (g(g+2)−(2g+1)H_g)/(g−1), where H_g=∑_k=1^g 1/k denotes the g-th harmonic number. If X is hyperelliptic, we give the stronger bound φ(X)>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≥ 2 over a number field and hence new uniform height bounds in the Bogomolov conjecture.
ER  -