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Non-archimedean Arakelov theory. (English) Zbl 0866.14011
This lengthy paper deals with a non-Archimedean analogue of Arakelov theory. While Arakelov theory starts with a variety $$X$$ over a number field (say $$\mathbb{Q})$$ and tries to add data to the usual archimedean fibres $$X(\mathbb{C})$$ so that one obtains Chow rings and intersection pairings one can go back and try to see what one gets if one mimicks the Archimedean constructions at the finite fibres. S. Zhang obtained interesting results in his paper ‘Adimissible pairing on a curve’ [Invent. Math. 112, No. 1, 171-193 (1993; Zbl 0795.14015)] constructing the Arakelov metric at the finite places. Here the authors are planning a bigger enterprise and set up a number of basic ingredients. They define the analogs of the groups of closed $$C^\infty$$ forms of type $$(p,p)$$ and the group of $$C^\infty$$ forms of type $$(p,p)$$ modulo the image of $$\partial$$ and $$\overline \partial$$ as certain limits over regular models. They prove a regularity result for such forms and introduce the groups $$CH^p$$ and prove that such groups can be seen as a limit over models. Then they extend some well-known results from the classical case to the new setting. In the last section they deal with the case of curves and obtain results close to those of Zhang. The authors point out that a non-Archimedean analogue of the arithmetic Riemann-Roch theorem is still lacking. There is an appendix on operational Chow-cohomology.

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights 14G20 Local ground fields in algebraic geometry