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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 is a hermitian vector bundle on an arithmetic scheme X, its arithmetic Atiyah class b at X/Z (E) lies in the group d Ext 2000 Mathematics Subject Classification. -MSC: Primary 14G40; Secondary 11J95, 14F05, 32L10.

Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L an , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L an and that the non-archimedean Monge-Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.