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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 .
The analytic Deligne complex Z(p)<sub>D, an</sub> on a complex analytic manifold X is<blockquote></blockquote>where Z(p) = (2ài)<sup>p</sup>Z. Depending on the context, is either the complex of smooth (i.e., C<sup>âÂÂ</sup>) 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<blockquote></blockquote>
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<sup>×</sup>-bundles over X. For p = q = 2, it is the group of isomorphism classes of C<sup>×</sup>-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 ().
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<blockquote> </blockquote>
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 where for odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.