Deligne cohomology
In mathematics, Deligne cohomology sometimes called Deligne-Beilinson cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.
For introductory accounts of Deligne cohomology see,, and.
Definition
The analytic Deligne complex ZD, an on a complex analytic manifold X iswhere Z = pZ. Depending on the context, is either the complex of smooth differential forms or of holomorphic forms, respectively.
The Deligne cohomology is the q-th hypercohomology of the Deligne complex. An alternative definition of this complex is given as the homotopy limit of the diagram
Properties
Deligne cohomology groups can be described geometrically, especially in low degrees. For p = 0, it agrees with the q-th singular cohomology group, by definition. For q = 2 and p = 1, it is isomorphic to the group of isomorphism classes of smooth principal C×-bundles over X. For p = q = 2, it is the group of isomorphism classes of C×-bundles with connection. For q = 3 and p = 2 or 3, descriptions in terms of gerbes are available. This has been generalized to a description in higher degrees in terms of iterated classifying spaces and connections on them.
Relation with Hodge classes
Recall there is a subgroup of integral cohomology classes in called the group of Hodge classes. There is an exact sequence relating Deligne-cohomology, their intermediate Jacobians, and this group of Hodge classes as a short exact sequence
Applications
Deligne cohomology is used to formulate Beilinson conjectures on special values of L-functions.
Extensions
There is an extension of Deligne-cohomology defined for any symmetric spectrum where for odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.