Zhang’s conjectural lower bound for the admissible canonical self-intersection

Establish Zhang’s conjectured inequality relating the self-intersection number of the admissible adelic metrized canonical bundle of a smooth projective curve to the Zhang–Kawazumi invariants of its polarized metrized reduction graphs and complex fibers: for a smooth projective curve C of genus g≥2 over a number field K with semistable model B, prove that ω_a^2 is at least (2g−2)/(2g+1) times the sum of the non-Archimedean contributions φ(Γ_v(C)) log N(v) and the Archimedean contributions φ(C_σ).

Background

For a smooth projective curve C of genus g≥2 over a number field K, Zhang’s inequality conjectures a lower bound for the arithmetic self-intersection number ωa^2 of the admissible adelic metrized canonical bundle. The proposed lower bound combines local contributions from the polarized metrized reduction graphs Γ_v(C) at finite places with complex-analytic Zhang–Kawazumi invariants φ(Cσ) at embeddings σ:K→ℂ.

The paper notes that Zhang’s inequality follows from the more general arithmetic Standard conjecture of Gillet–Soulé and that equality is known for hyperelliptic curves. The results proved in the paper establish different explicit lower bounds for ω_a^2 by combining an existing inequality with new uniform lower bounds for the Zhang–Kawazumi invariant; they do not prove Zhang’s conjectured inequality itself.