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.


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


Zbl 0741.14012