Deligne cohomology
# Deligne cohomology

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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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

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

where Z(p) = (2π $i)^pZ.$ Depending on the context, $\Omega _{X}^{*}$ 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 of the diagram

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

## 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 ${\text{Hdg}}^{p}(X)\subset H^{p,p}(X)$ of integral cohomology classes in $H^{2p}(X)$ 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

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

## 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 $E$ where $\pi _{i}(E)\otimes \mathbb {C} =0$ for $i$ odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.

