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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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" isformula_1where Z("p") = (2π i)"p"Z. Depending on the context, formula_2 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. An alternative definition of this complex is given as the homotopy limit of the diagramformula_3
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 ().
Relation with Hodge classes.
Recall there is a subgroup formula_4 of integral cohomology classes in formula_5 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 sequenceformula_6 
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 formula_7 where formula_8 for formula_9 odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.