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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.

For introductory accounts of Deligne cohomology see Brylinski (2008, section 1.5), Esnault & Viehweg (1988), and Gomi (2009, section 2).

Definition

The analytic Deligne complex Z(p)_{D, an} on a complex analytic manifold X is

${\displaystyle 0\rightarrow \mathbf {Z} (p)\rightarrow \Omega _{X}^{0}\rightarrow \Omega _{X}^{1}\rightarrow \cdots \rightarrow \Omega _{X}^{p-1}\rightarrow 0\rightarrow \dots }$

where Z(p) = (2π i)^pZ. Depending on the context, ${\displaystyle \Omega _{X}^{*))$ 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. An alternative definition of this complex is given as the homotopy limit^[1] of the diagram

${\displaystyle {\begin{matrix}&&\mathbb {Z} \\&&\downarrow \\\Omega _{X}^{\bullet \geq p}&\to &\Omega _{X}^{\bullet }\end{matrix))}$

Properties

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)).

Relation with Hodge classes

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, their intermediate Jacobians, and this group of Hodge classes as a short exact sequence

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

Applications

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

Extensions

There is an extension of Deligne-cohomology defined for any symmetric 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.