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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 ZD, an on a complex analytic manifold X is
where 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.

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.

Applications

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