The goal of Arakelov Theory is to produce (deep) arithmetic results by combining tools from algebraic geometry over Spec \(\mathbb Z\) with those from complex geometry. Indeed, it was first invented with the hope of proving the Mordell conjecture on the finiteness of the set of rational points on a curve of genus at least two over a number field.
(This dream was eventually realized by G. Faltings [Invent. Math. 73, 349–366 (1983; Zbl 0588.14026)].) When interpreted with hindsight, the simplest example of the Arakelov philosophy is Minkowski’s geometry of numbers, which has been described beautifully by L. Szpiro [Astérisque 127, 11–28 (1985; Zbl 1182.11029)].
In the paper under review, the authors begin a sequence of investigations of arithmetic extension groups. Let \(X\) be an arithmetic scheme — i.e., a separated scheme that is flat and of finite type over Spec \(\mathbb Z\), and such that the associated complex variety \(X_{\mathbb C}\) is smooth. If \(F\) and \(G\) are locally free coherent \({\mathcal O}_X\)-modules (vector bundles on \(X\)), an arithmetic extension \(({\mathcal E}, s)\) of \(F\) by \(G\) is an extension of \({\mathcal O}_X\)-modules \({\mathcal E}: 0 \to G \to E \to F \to 0\) together with a complex-conjugation-invariant \({\mathcal C}^\infty\)-splitting \(s: F_{\mathbb C} \to E_{\mathbb C}\) of the associated extension of complex vector bundles over the manifold of complex points \(X(\mathbb C)\). The set of isomorphism classes of arithmetic extensions, denoted \(\widehat{\text{Ext}}^1_X(F, G)\), becomes an Abelian group when equipped with the Baer sum operation.
Arithmetic extensions arise naturally in Arakelov geometry via admissible extensions: if \(\overline{F}\) and \(\overline{G}\) are Hermitian vector bundles on \(X\), then an admissible extension of \(\overline{F}\) by \(\overline{G}\) is an exact sequence of Hermitian vector bundles \(\overline{{\mathcal E}}: 0 \to \overline{G} \to \overline{E} \to \overline{F} \to 0\) for which the metrics on \(\overline{F}\) and \(\overline{G}\) agree with the quotient and subspace metrics induced by \(\overline{E}\), respectively. The authors observe that, in this context, isomorphism classes of arithmetic extensions of \(F\) by \(G\) are in bijective correspondence with isomorphism classes of admissible extensions of \(\overline{F}\) by \(\overline{G}\). As an important special case, if \({\mathcal O}_K\) is the ring of integers in a number field \(K\) and \(X = \text{Spec} \, {\mathcal O}_K\) is the associated arithmetic curve, then any locally free coherent \({\mathcal O}_X\)-module can be endowed with a Hermitian metric (by choosing a basis for the associated complex vector space), and hence every arithmetic extension of \({\mathcal O}_X\)-modules arises as an admissible extension.
The authors explain how the group of arithmetic extensions \(\widehat{\text{Ext}}^1_X(F, G)\) itself arises as an extension of the “classical extension group” \(\text{Ext}^1_{{\mathcal O}_X}(F, G)\) by a group of analytic type. They give various functorial properties of this group, and discuss the relationship between their notion of arithmetic extension and the “arithmetic torsors” of A. Chambert-Loir and Y. Tschinkel [Prog. Math. 199, 37–70 (2001; Zbl 1001.14005)]. The later part of the paper is devoted to studying slopes of arithmetic extensions over arithmetic curves. A number of classical results in the geometry of numbers and the reduction theory of quadratic forms are examined in this framework. For example, Theorem 4.3.1 exhibits the phenomenon of “bounded distortion” for the splitting of lattices: any \({\mathcal O}_K\)-lattice in \(\mathbb C^n\) differs from a rectangular lattice by a bounded amount, depending only on \(K\) and \(n\).
The authors plan to extend their results in two subsequent papers to discuss the arithmetic Atiyah extension, the arithmetic Hodge extension, and the arithmetic Schwarz extension. A sketch of these ideas is given in the introduction to the present paper.