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Arithmetic Chow rings and Deligne-Beilinson cohomology. (English) Zbl 0922.14002
The arithmetic Chow ring of a complex projective manifold has been introduced by Gillet-Soulé as an attempt to generalize to higher dimensions the Arakelov intersection theory [see H. Gillet and C. Soulé, Publ. Math., Inst. Hautes Étud. Sci. 72, 93-174 (1990; Zbl 0741.14012)]. In this theory one associates to every algebraic cycle a Green current and a $$*$$-product of Green currents for any two algebraic cycles intersecting properly. This $$*$$-product corresponds to the intersection of cycles. Gillet and Soulé also extended the definition of arithmetic Chow rings and in the projective case related the Green currents to Deligne-Beilinson cohomology. Thus, for the latter problem in the quasi-projective case they suggested that the definition of arithmetic Chow rings could be improved in order to relate it with Deligne-Beilinson cohomology as well. The aim of the paper under review is precisely to extend the definition of arithmetic Chow ring to the quasi-projective case so that a relation to Deligne-Beilinson cohomology be possible. To this end the author gives a new definition of the space of Green forms associated to algebraic cycles by using Deligne-Beilinson cohomology. This new definition is compatible with the previous one for projective manifolds.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14C25 Algebraic cycles 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies