##
**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.

Reviewer: Lucien Bădescu (Bucureşti)

### 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 |