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Successive minima and asymptotic slopes in Arakelov geometry

Part of: Geometry of numbers Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  10 June 2021

François Ballaÿ
François Ballaÿ*
Affiliation:
Beijing International Center for Mathematical Research, Peking University, 5 Yi He Yuan Road, Beijing100871, PR Chinafrancois.ballay@bicmr.pku.edu.cn
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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.

Keywords

heightessential minimumsuccessive minimaadelic line bundles and divisorsOkounkov bodieshermitian vector bundlestransference theorems

MSC classification

Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Secondary: 11G50: Heights 11H06: Lattices and convex bodies
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 6 , June 2021 , pp. 1302 - 1339
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Copyright
© The Author(s) 2021

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