Mathematik Hermitian vector bundles and extension groups on arithmetic schemes . II . The arithmetic Atiyah extension

— 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 is a hermitian vector bundle on an arithmetic scheme X, its arithmetic Atiyah class b atX/Z(E) lies in the group d Ext1X(E,E ⊗ ΩX/Z), and is an obstruction to the algebraicity over X of the unitary connection on the vector bundle EC 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 ĉ1 (L) of a hermitian line bundle L in the arithmetic Hodge cohomology group d Ext1X(OX ,ΩX/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 ĉ1 . 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 atX/CL in H 1(X,Ω1 X/C ). We describe precisely when atX/CL vanishes. In particular, when the fixed part of the relative Picard variety of X over C is trivial, this holds iff some positive power of L descends to a line bundle over C. 2000 Mathematics Subject Classification. — MSC: Primary 14G40; Secondary 11J95, 14F05, 32L10.