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.