Semipurity of tempered Deligne cohomology

TL;DR

This paper introduces formal and tempered Deligne cohomology groups, establishes a duality between them, and proves a semipurity property that simplifies the understanding of covariant arithmetic Chow groups for projective varieties.

Contribution

It defines new cohomology groups and proves a duality and semipurity property, advancing the study of covariant arithmetic Chow groups in algebraic geometry.

Findings

Established duality between formal and tempered Deligne cohomology

Proved vanishing theorem for formal Deligne cohomology

Demonstrated semipurity property for tempered Deligne cohomology

Abstract

In this paper we define 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. We then prove the existence of a duality between them, a vanishing theorem for the former and a semipurity property for the latter. The motivation of these results comes from the study of covariant arithmetic Chow groups. The semi-purity property of tempered Deligne cohomology implies, in particular, that several definitions of covariant arithmetic Chow groups agree for projective arithmetic varieties.