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Direct images in non-archimedean Arakelov theory. (English) Zbl 0969.14015
An earlier paper by S. Bloch, H. Gillet, and C. Soulé [J. Algebr. Geom. 4, No. 3, 427-485 (1995; Zbl 0866.14011)] constructed a theory for a variety \(X\) over a number field that does at finite places what the Arakelov-Gillet-Soulé intersection theory does at infinite places of the number field. For example, this earlier paper found non-Archimedean analogs of the groups of \(C^\infty\) forms on \(X(\mathbb C_v)\); these are central to Gillet-Soulé intersection theory at infinite places.
The present paper fills in one notable omission in that earlier paper: an analogue of the Riemann-Roch-Grothendieck theorem in this context. The starting point is the definition of a direct image that respects base change and change of model. For this definition, the authors then define Grothendieck groups and higher analytic torsion currents, leading up to a Riemann-Roch theorem for this direct image.
Reviewer: P.Vojta (Berkeley)

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C40 Riemann-Roch theorems
14G20 Local ground fields in algebraic geometry
14C15 (Equivariant) Chow groups and rings; motives
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