Deligne cohomology

In mathematics, Deligne cohomology sometimes called Deligne-Beilinson 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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 Template:Harvtxt, Template:Harvtxt, and Template:Harvtxt.

Definition

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

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

where Z(p) = (2π i)^pZ. Depending on the context,

${\mathrm{\Omega }}_X^{*}$

is either the complex of smooth (i.e., C^∞) differential forms or of holomorphic forms, respectively. The Deligne cohomology Template:Nowrap is the q-th hypercohomology of the Deligne complex. An alternative definition of this complex is given as the homotopy limit of the diagram

$\begin{array}{ccc}& & \mathbb{Z}\\ & & \downarrow \\ {\mathrm{\Omega }}_X^{\bullet \ge p}& \to & {\mathrm{\Omega }}_X^{\bullet }\end{array}$

Properties

Deligne cohomology groups Template:Nowrap 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 (Template:Harvtxt). This has been generalized to a description in higher degrees in terms of iterated classifying spaces and connections on them (Template:Harvtxt).

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

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

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

See also

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

References

• Template:Cite web
• Geometry of Deligne cohomology
• Notes on differential cohomology and gerbes
• Twisted smooth Deligne cohomology
• Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation