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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.
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
D,an (X,Z(p)) is the q-th hypercohomology of the Deligne complex.
Deligne cohomology groups H q
D (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 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)).
- Brylinski, Jean-Luc (2008) , Loop spaces, characteristic classes and geometric quantization, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4731-5, ISBN 978-0-8176-4730-8, MR 2362847
- Esnault, Hélène; Viehweg, Eckart (1988), "Deligne-Beĭlinson cohomology" (PDF), Beĭlinson's conjectures on special values of L-functions, Perspect. Math. 4, Boston, MA: Academic Press, pp. 43–91, ISBN 978-0-12-581120-0, MR 944991
- Gajer, Pawel (1997), "Geometry of Deligne cohomology", Inventiones Mathematicae 127 (1): 155–207, doi:10.1007/s002220050118, ISSN 0020-9910
- Gomi, Kiyonori (2009), "Projective unitary representations of smooth Deligne cohomology groups", Journal of Geometry and Physics 59 (9): 1339–1356, arXiv:math/0510187, doi:10.1016/j.geomphys.2009.06.012, ISSN 0393-0440, MR 2541824