Line bundles, connections, Deligne-Beilinson and absolute Hodge cohomology

TL;DR

This paper explores the relationship between Picard groups, line bundles, and various cohomology theories, proposing modifications to Deligne-Beilinson cohomology to account for irregular connections and linking it to absolute Hodge cohomology.

Contribution

It introduces a modified Deligne-Beilinson cohomology framework and an intermediate object, connecting line bundles with regular and irregular connections to cohomology theories.

Findings

Picard group can be expressed by absolute Hodge cohomology.

Modified Deligne-Beilinson cohomology accounts for irregular connections.

An intermediate object bridges Picard group and Deligne-Beilinson cohomology.

Abstract

It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not true, as already remarked by M. Saito. We explain how one has to modify the latter, show that the Picard group can be expressed by absolute Hodge cohomology, too, and introduce an intermediate object between Picard group and usual Deligne-Beilinson cohomology group. Similarly as in the case of Deligne cohomology one can relate line bundles with a regular connection to (modified) Deligne-Beilinson cohomology. In order to take irregular connections into account one has to change the definition of Deligne-Beilinson cohomology even more.