Английская Википедия: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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 Шаблон:Harvtxt, Шаблон:Harvtxt, and Шаблон:Harvtxt.
Definition
The analytic Deligne complex Z(p)D, an on a complex analytic manifold X is
<math>0\rightarrow \mathbf Z(p)\rightarrow \Omega^0_X\rightarrow \Omega^1_X\rightarrow\cdots\rightarrow \Omega_X^{p-1} \rightarrow 0 \rightarrow \dots</math>
where Z(p) = (2π i)pZ. Depending on the context, <math>\Omega^*_X</math> is either the complex of smooth (i.e., C∞) differential forms or of holomorphic forms, respectively. The Deligne cohomology Шаблон:Nowrap 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
<math>\begin{matrix}
& & \mathbb{Z} \\ & & \downarrow \\ \Omega_X^{ \bullet \geq p} & \to & \Omega_X^\bullet
\end{matrix}</math>
Properties
Deligne cohomology groups Шаблон: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 (Шаблон:Harvtxt). This has been generalized to a description in higher degrees in terms of iterated classifying spaces and connections on them (Шаблон:Harvtxt).
Relation with Hodge classes
Recall there is a subgroup <math>\text{Hdg}^p(X) \subset H^{p,p}(X)</math> of integral cohomology classes in <math>H^{2p}(X)</math> 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
<math>0 \to J^{2p-1}(X) \to H^{2p}_\mathcal{D}(X,\mathbb{Z}(p)) \to \text{Hdg}^{2p}(X) \to 0</math>
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 <math>E</math>[1] where <math>\pi_i(E)\otimes \mathbb{C} = 0</math> for <math>i</math> odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.
See also
References
- Deligne-Beilinson cohomology
- Geometry of Deligne cohomology
- Notes on differential cohomology and gerbes
- Twisted smooth Deligne cohomology
- Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
