Deligne cohomology explained

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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 → Z(p) → \Omega ^{0} _{X →} \Omega ^{1} _{X → … →} \Omega ^{p -1} _{X} → 0 → ...

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 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

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

Properties

Deligne cohomology groups 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 . 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

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 ^{2 p} _{l{D}(X, Z (p)) \to} Hdg ^{2p} (X) \to 0

Applications

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

Extensions

E where \pi _{i(E) ⊗} 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

• Web site: Deligne-Beilinson cohomology. https://web.archive.org/web/20200603081515/http://page.mi.fu-berlin.de/esnault/preprints/ec/deligne_beilinson.pdf . 2020-06-03 . dead .
• Geometry of Deligne cohomology
• Notes on differential cohomology and gerbes
• Twisted smooth Deligne cohomology
• Bloch's Conjecture, Deligne Cohomology and Higher Chow Groups

Notes and References

1. Hopkins. Michael J.. Quick. Gereon. March 2015. Hodge filtered complex bordism. Journal of Topology. 8. 1. 147–183. 10.1112/jtopol/jtu021. 1212.2173. 16757713 .