Arithmetic intersection theory. (English) Zbl 0741.14012

Gillet, Henri; Soulé, Christophe

Publ. Math., Inst. Hautes Étud. Sci. 72, 93-174 (1990).

The article constructs an intersection-product on the arithmetic Chow groups \(\widehat{Ch}\) of regular schemes over the integers of number- fields. These groups are quotients of pairs \((Z,g_ Z)\), \(Z\) a cycle and \(g_ Z\) a “Green’s current”, under rational equivalence. The main two difficulties are first to define the product of Green’s currents (which needs some regularity-considerations) and second an analytic version of Chow’s moving lemma. It should be noted that, as in the purely algebraic case, the theory becomes much simpler if one only attempts to intersect with Chern-classes of hermitian bundles (these have been defined earlier by the same authors).

Reviewer: G.Faltings

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14C25	Algebraic cycles

Keywords:

intersection-product on the arithmetic Chow groups; Green’s currents; Chow’s moving lemma

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