Minima, pentes et alg\`ebre tensorielle

TL;DR

This paper explores the properties of slopes and minima of adelic vector bundles, establishing inequalities and bounds for tensor products and powers, and extending classical results like Minkowski-Hlawka and Siegel lemma in this context.

Contribution

It introduces new bounds for tensor products and exterior/symmetric powers of hermitian bundles, improving previous estimates and connecting minima with slopes in adelic vector bundle theory.

Findings

Proves a Minkowski-Hlawka type theorem for adelic vector bundles.

Provides bounds for absolute minima and maximal slopes of tensor products.

Extends inequalities to exterior and symmetric powers involving multinomial coefficients.

Abstract

Slopes of an adelic vector bundle exhibit a behaviour akin to successive minima. Comparisons between the two amount to a Siegel lemma. Here we use Zhang's version for absolute minima over the algebraic numbers. We prove a Minkowski-Hlawka theorem in this context. We also study the tensor product of two hermitian bundles bounding both its absolute minimum and maximal slope, thus improving an estimate of Chen. We further include similar inequalities for exterior and symmetric powers, in terms of some lcm of multinomial coefficients.