Successive minima and asymptotic slopes in Arakelov geometry

Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\bar {D}$ be an adelic ${\mathbb {R}}$ -Cartier divisor on $X$ . We prove a conjecture of Chen, showing that the essential minimum $\zeta _{\mathrm {ess}}(\bar {D})$ of $\bar {D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\zeta _{\mathrm {ess}}(\bar {D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom–Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = {\mathbb {P}}_K^{d}$ , our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and Rémond.

Arakelov geometry on degenerating curves

We study the behaviour of the Arakelov metric on a smooth curve under semistable degeneration. The final result is a complicated formula involving the local discriminants of the singularities, and the graph governing the degeneration.