Direct Images in Non-archimedean Arakelov Theory

In this paper we develop a formalism of direct images for metrized vector bundles in the context of the non-archimedean Arakelov theory introduced in our joint work [BGS] with S. Bloch, and we prove a Riemann-Roch-Grothendieck theorem for this direct image. The new ingredient in the construction of the direct image is a non archimedean " analytic torsion current ". Let K be the fraction field of a discrete valuation ring Λ, and X a smooth projective variety over K. In [BGS] we defined the codimension p arithmetic Chow group of X as the inductive limit CH p (X) = lim −→ CH p (X) of the Chow groups of the models X of X over Λ. Assuming resolution of singularities (cf. 1.1 below) we proved that these groups can also be defined as rational equivalence classes of pairs (Z, g), where Z is a codimension p cycle on X, and g is a " Green current " for Z. Here a " current " is a projective system of cycle classes on the special fibers of all possible models of X. We have shown in [BGS] that many concepts and results in complex geometry and arithmetic intersection theory [GS1] have analogs in this context: differential forms, ∂∂-lemma, Poincaré-Lelong formula, intersection product, inverse and direct image maps in arithmetic Chow groups etc. On the other hand, we defined a metrized vector bundle on X to be a bundle E on X, together with a bundle E X on some model X of X which restricts to E on X. The theory of characteristic classes (resp. Bott-Chern secondary characteristic classes) for hermitian vector bundles on arithmetic varieties [GS2] is replaced here by characteristic classes with values in the Chow groups of X (resp. the Chow groups of X with supports in its special fiber) ([BGS], (1.9), and § 2 below). These classes are contravariant for maps of varieties over K. However, we were not able in [BGS] to define direct images of metrized vector bundles. Recall that in Arakelov geometry, if f : X → Y is a map of varieties over Z which is smooth on the set of complex points of X, and if E is an hermitian vector bundle on X, once we choose a metric on T f , the L 2-metric on the determinant line bundle det(Rf * E) need not be smooth in general. For …