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Slopes of adelic vector bundles over global fields. (Pentes des fibrés vectoriels adéliques sur un corps global.) (French) Zbl 1206.14047
This paper is a generalization of Bost’s slope theory for Hermitian vector bundles over number fields, whose importance is the application to Diophantine approximation, known as the slope method. The adelic approach in this paper provides a generalization to all global fields, which recovers Bost’s theory in the case of number fields. A possible application is seen in a slope method for function fields.
The underlying definition of the theory developed in this paper is that of an adelic vector bundle over a global field \(k\), which is vector space \(E\) over \(k\) together with a \(^{Gal}(\mathbb{C}_v,k_v)\)-invariant norm \(||\cdot ||_v:E\otimes_k \mathbb{C}_v \to \mathbb{R}_{\geq0}\) for every place \(v\) of \(k\). Here, \(\mathbb{C}_v\) denotes the completion of the algebraic closure of \(k_v\).
After an introduction, the author reviews in Section 2 some classical methods: Minkowski theory, the ellipsoids of maximal and minimal volume (“John ellipsoids” resp. “Löwner ellipsoids”), quotient volumes, the Banach-Mazur distance of two convex sets, direct sums and tensor products of normed vector spaces.
In Section 3, the author introduces adelic vector bundles and explains how to connect them to Hermitian vector bundles. He provides basic constructions as subvectorbundles, quotients, direct sums, duals, operator norms, tensor products, exterior products, determinants and symmetric products.
In Section 4, Gaudron defines the adelic degree of an adelic vector bundle \((E,\{||\cdot||_v\})\), which is the logarithmic difference of the volume of the unit ball in \(E\) with respect to all local norms \(||\cdot ||_v\) and the volume of the unit ball with respect to the quadratic norm \(||\cdot||_2\) for some choosen basis. He introduces John and Löwner vector bundles, respectively, in the adelic context and investigates their adelic degrees. He eplains how the adelic degree behaves with respect to extensions of scalars, direct sums, duals, quotients and sums of embedded adelic vector bundles.
In section 5, he defines the slope of an adelic vector bundle, together with some other invariants. Most important among these is the maximal slope, whose existence is proved in the first part of this section. Gaudron reviews the canonical filtration of a Hermitian vector bundle from the adelic view point and discusses semi-stable vector bundles.
In section 6, the author compares the slopes of adelic vector bundles in terms of morphisms between them.
In section 7, he proves that the maximal slope of the \(l\)-th symmetric power of an adelic vector bundle \(E\) equals \(l\) times the maximal slope of \(E\).
Finally, in Section 8, Gaudron explains some possible applications to geometry.

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11R56 Adèle rings and groups
14D20 Algebraic moduli problems, moduli of vector bundles
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