Successive minima and asymptotic slopes in Arakelov geometry
Résumé
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.
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