Bost, Jean-Benoit and Kuennemann, Klaus (2009) HERMITIAN VECTOR BUNDLES AND EXTENSION GROUPS ON ARITHMETIC SCHEMES II. THE ARITHMETIC ATIYAH EXTENSION. ASTERISQUE (327). pp. 361-424. ISSN 0303-1179,
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In a previous paper, we have defined arithmetic extension groups in the context of Arakelov geometry. In the present one, we introduce an arithmetic analogue of the Atiyah extension that defines an element - the arithmetic Atiyah class - in a suitable arithmetic extension group. Namely, if (E) over bar is a hermitian vector bundle on an arithmetic scheme X, its arithmetic Atiyah class (at) over cap (X/Z)((E) over bar) lies in the group (Ext) over cap (1)(X) (E, E circle times Omega(1)(X/Z)), and is an obstruction to the algebraicity over X of the unitary connection on the vector bundle E-C over the complex manifold X(C) that is compatible with its holomorphic structure. In the first sections of this article, we develop the basic properties of the arithmetic Atiyah class which can be used to define characteristic classes in arithmetic Hodge cohomology. Then we study the vanishing of the first Chern class (c) over cap (H)(1)((L) over bar) of a hermitian line bundle L in the arithmetic Hodge cohomology group (Ext) over cap (1)(X) (O-X, Omega(1)(X/Z)). This may be translated into a concrete problem of diophantine geometry, concerning rational points of the universal vector extension of the Picard variety of X. We investigate this problem, which was already considered and solved in some cases by Bertrand, by using a classical transcendence result of Schneider-Lang, and we derive a finiteness result for the kernel of (c) over cap (H)(1). In the final section, we consider a geometric analog of our arithmetic situation, namely a smooth, projective variety X which is fibered on a curve C defined over some field k of characteristic zero. To any line bundle L over X is attached its relative Atiyah class at(X/C)L in H-1(X, Omega(1)(X/C)). We describe precisely when at(X/C)L vanishes. In particular, when the fixed part of the relative Picard variety of X over C is trivial, this holds if some positive power of L descends to a line bundle over C.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | POINTS; TRACE; Arakelov geometry; hermitian vector bundles; extension groups; Atiyah class; transcendence and algebraic groups |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Klaus Künnemann |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 12 Oct 2020 05:17 |
| Last Modified: | 12 Oct 2020 05:17 |
| URI: | https://pred.uni-regensburg.de/id/eprint/29645 |
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