Inmathematics,Deligne cohomology sometimes calledDeligne-Beilinson cohomology is thehypercohomology of theDeligne complex of acomplex manifold. It was introduced byPierre Deligne in unpublished work in about 1972 as a cohomology theory foralgebraic varieties that includes both ordinary cohomology andintermediate Jacobians.

For introductory accounts of Deligne cohomology seeBrylinski (2008, section 1.5),Esnault & Viehweg (1988), andGomi (2009, section 2).

The analytic Deligne complexZ(p)_{D, an} on a complex analytic manifoldX is

${\displaystyle 0\to \mathbf{Z}(p)\to {\mathrm{\Omega }}_X^0\to {\mathrm{\Omega }}_X^1\to \cdots \to {\mathrm{\Omega }}_X^{p-1}\to 0\to \dots }$

whereZ(p) = (2π i)^pZ. Depending on the context,${\displaystyle {\mathrm{\Omega }}_X^{\ast }}$ is either the complex of smooth (i.e.,C^∞)differential forms or of holomorphic forms, respectively.The Deligne cohomologyH q
D,an (X,Z(p)) is theq-th hypercohomology of the Deligne complex. An alternative definition of this complex is given as the homotopy limit^[1] of the diagram

Deligne cohomology groupsH q
D (X,Z(p)) can be described geometrically, especially in low degrees. Forp = 0, it agrees with theq-th singular cohomology group (withZ-coefficients), by definition. Forq = 2 andp = 1, it is isomorphic to the group of isomorphism classes of smooth (or holomorphic, depending on the context)principalC^×-bundles overX. Forp =q = 2, it is the group of isomorphism classes ofC^×-bundles withconnection. Forq = 3 andp = 2 or 3, descriptions in terms ofgerbes are available (Brylinski (2008)). This has been generalized to a description in higher degrees in terms of iteratedclassifying spaces and connections on them (Gajer (1997)).

Recall there is a subgroup${\displaystyle {\text{Hdg}}^p(X)\subset H^{p,p}(X)}$ of integral cohomology classes in${\displaystyle H^{2p}(X)}$ called the group of Hodge classes. There is an exact sequence relating Deligne-cohomology, theirintermediate Jacobians, and this group of Hodge classes as a short exact sequence

${\displaystyle 0\to J^{2p-1}(X)\to H_{\mathcal{D}}^{2p}(X,\mathbb{Z}(p))\to {\text{Hdg}}^{2p}(X)\to 0}$

Deligne cohomology is used to formulateBeilinson conjectures onspecial values of L-functions.

There is an extension of Deligne-cohomology defined for anysymmetric spectrum${\displaystyle E}$^[1] where${\displaystyle {\pi }_i(E)\otimes \mathbb{C}=0}$ for${\displaystyle i}$ odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.

• Bundle gerbe
• Motivic cohomology
• Hodge structure
• Intermediate Jacobian

• "Deligne-Beilinson cohomology"(PDF). Archived fromthe original(PDF) on 2020-06-03.
• Geometry of Deligne cohomology
• Notes on differential cohomology and gerbes
• Twisted smooth Deligne cohomology
• Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups
• Brylinski, Jean-Luc (2008) [1993],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., vol. 4, Boston, MA:Academic Press, pp. 43–91,ISBN 978-0-12-581120-0,MR 0944991
• Gajer, Pawel (1997), "Geometry of Deligne cohomology",Inventiones Mathematicae,127 (1):155–207,arXiv:alg-geom/9601025,Bibcode:1996InMat.127..155G,doi:10.1007/s002220050118,ISSN 0020-9910,S2CID 18446635
• Gomi, Kiyonori (2009), "Projective unitary representations of smooth Deligne cohomology groups",Journal of Geometry and Physics,59 (9):1339–1356,arXiv:math/0510187,Bibcode:2009JGP....59.1339G,doi:10.1016/j.geomphys.2009.06.012,ISSN 0393-0440,MR 2541824,S2CID 17437631

• Sheaf theory
• Homological algebra
• Cohomology theories