Deligne cohomology

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne (1971) 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).


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 H q
is the q-th hypercohomology of the Deligne complex.


Deligne cohomology groups H q
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 (Brylinski (2008)). This has been generalized to a description in higher degrees in terms of iterated classifying spaces and connections on them (Gajer (1997)).


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