Deligne cohomology
In mathematics, Deligne 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 It was introduced by Pierre Deligne in unpublished work in about 1972  as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 Z(p)D, an on a complex analytic manifold X is

where Z(p) = (2π i)pZ. Depending on the context,  is either the complex of smooth (i.e., C∞) differential forms or of holomorphic forms, respectively.
The Deligne cohomology  is the q-th hypercohomology of the Deligne complex.

Properties

Deligne cohomology groups  can be described geometrically, especially in low degrees. For p = 0, it agrees with the q-th singular cohomology group (with Z-coefficients), by definition. For q = 2 and p = 1, it is isomorphic to the group of isomorphism classes of smooth (or holomorphic, depending on the context) 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 ().

Applications
Deligne cohomology is used to formulate Beilinson conjectures on special values of L-functions.