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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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

Definition

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

[math]\displaystyle{ 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]\displaystyle{ \Omega^*_X }[/math] 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 of the diagram

[math]\displaystyle{ \begin{matrix} & & \mathbb{Z} \\ & & \downarrow \\ \Omega_X^{ \bullet \geq p} & \to & \Omega_X^\bullet \end{matrix} }[/math]

Properties

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

Recall there is a subgroup [math]\displaystyle{ \text{Hdg}^p(X) \subset H^{p,p}(X) }[/math] of integral cohomology classes in [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ E }[/math] where [math]\displaystyle{ \pi_i(E)\otimes \mathbb{C} = 0 }[/math] for [math]\displaystyle{ i }[/math] odd which can be compared with ordinary Deligne cohomology on complex analytic varieties.

See also

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

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
• Brylinski, Jean-Luc (2008), Loop spaces, characteristic classes and geometric quantization, Modern Birkhäuser Classics, Boston, MA: Birkhäuser Boston, doi:10.1007/978-0-8176-4731-5, ISBN 978-0-8176-4730-8 
• Esnault, Hélène; Viehweg, Eckart (1988), "Deligne-Beĭlinson cohomology", Beĭlinson's conjectures on special values of L-functions, Perspect. Math., 4, Boston, MA: Academic Press, pp. 43–91, ISBN 978-0-12-581120-0, http://www.uni-due.de/~mat903/preprints/ec/deligne_beilinson.pdf 
• Gajer, Pawel (1997), "Geometry of Deligne cohomology", Inventiones Mathematicae 127 (1): 155–207, doi:10.1007/s002220050118, ISSN 0020-9910, Bibcode: 1996InMat.127..155G 
• Gomi, Kiyonori (2009), "Projective unitary representations of smooth Deligne cohomology groups", Journal of Geometry and Physics 59 (9): 1339–1356, doi:10.1016/j.geomphys.2009.06.012, ISSN 0393-0440, Bibcode: 2009JGP....59.1339G