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An arithmetic Riemann-Roch theorem. (English) Zbl 0777.14008
This paper contains full details of the fundamental results announced in two previous papers [see the authors, C. R. Acad. Sci., Paris, Sér. I 309, No. 17, 929-932 (1989; Zbl 0732.14002) and H. Gillet in Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 403-412 (1991; see the preceding review)]. Given an arithmetic variety \(X\) together with a Hermitian metric on its set of complex points, the “absolute” arithmetic Riemann-Roch theorem computes the degree (in the sense of Arakelov) of the determinant of the cohomology of a Hermitian vector bundle on \(X\) equipped with the Quillen metric. The computation is in terms of characteristic classes, which are invariants of the Hermitian bundle and \(X\), in the arithmetic Chow ring of \(X\). Actually, as in the case of the usual Grothendieck-Riemann-Roch theorem in algebraic geometry, the authors state and prove the “relative” version of this theorem. They use some ideas of Arakelov, Faltings and Deligne when they develop all these fundamental arithmetic concepts in arbitrary dimension.

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C40 Riemann-Roch theorems
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