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It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 Brylinski (2008, section 1.5), Esnault & Viehweg (1988), and Gomi (2009, section 2).
An alternative definition of this complex is given as the homotopy limit[1] of the diagram
Deligne cohomology groups H qD (X,Z(p)) 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 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 (Brylinski (2008)).
This has been generalized to a description in higher degrees in terms of iterated classifying spaces and connections on them (Gajer (1997)).
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
Deligne cohomology is used to formulate Beilinson conjectures on special values of L-functions.
There is an extension of Deligne-cohomology defined for any symmetric spectrum
odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.