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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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

Definition
The analytic Deligne complex Z(p)D, an on a complex analytic manifold X is"$0\rightarrow \mathbf Z(p)\rightarrow \Omega^0_X\rightarrow \Omega^1_X\rightarrow\cdots\rightarrow \Omega_X^{p-1} \rightarrow 0 \rightarrow \dots$"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 $H D,an q$(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 $H D q$(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&times;-bundles over X. For p = q = 2, it is the group of isomorphism classes of C&times;-bundles with connection. For q = 3 and p = 2 or 3, descriptions in terms of gerbes are available. This has been generalized to a description in higher degrees in terms of iterated classifying spaces and connections on them.

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

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.