Successive minima and asymptotic slopes in Arakelov geometry

Abstract

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.