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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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 Brylinski (2008, section 1.5), Esnault & Viehweg (1988), and Gomi (2009, section 2).

Definition

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

where Z(p) = (2π i)pZ. Depending on the context, 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[1] of the diagram

Properties

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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

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Recall there is a subgroup of integral cohomology classes in 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

Applications

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

Extensions

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There is an extension of Deligne-cohomology defined for any symmetric spectrum [1] where for odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.

See also

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References

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  1. ^ a b Hopkins, Michael J.; Quick, Gereon (March 2015). "Hodge filtered complex bordism". Journal of Topology. 8 (1): 147–183. arXiv:1212.2173. doi:10.1112/jtopol/jtu021. S2CID 16757713.