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Semi-purity of tempered Deligne cohomology. (English) Zbl 1194.14030

Motivated by the study of covariant arithmetic Chow groups, the author defines the formal and tempered Deligne cohomology groups, that are obtained by applying the Deligne complex functor to the complexes of formal differential forms and tempered currents respectively. He then constructs a (Poincaré like) duality between formal and tempered Deligne cohomology, that induces a perfect pairing between the corresponding separated vector spaces. He gets also a vanishing result for formal Deligne cohomology, which implies a semi-purity property for tempered Deligne cohomology thanks to the previous duality.
As applications, the author gets that several definitions of covariant arithmetic Chow groups agree for projective arithmetic varieties.

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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