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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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

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 q
D,an (X,Z(p)) is the q-th hypercohomology of the Deligne complex.

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)).

Applications

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

References

• Brylinski, Jean-Luc (2008) [1993], 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, MR2362847 
• 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 9780125811200, MR944991, 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 
• Gomi, Kiyonori (2009), "Projective unitary representations of smooth Deligne cohomology groups", Journal of Geometry and Physics 59 (9): 1339–1356, arXiv:math/0510187, doi:10.1016/j.geomphys.2009.06.012, ISSN 0393-0440, MR2541824