• Source: 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 It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians.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


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



      0
      →

      Z

      (
      p
      )
      →

      Ω

      X


      0


      →

      Ω

      X


      1


      →
      ⋯
      →

      Ω

      X


      p
      −
      1


      →
      0
      →
      …


      {\displaystyle 0\rightarrow \mathbf {Z} (p)\rightarrow \Omega _{X}^{0}\rightarrow \Omega _{X}^{1}\rightarrow \cdots \rightarrow \Omega _{X}^{p-1}\rightarrow 0\rightarrow \dots }

      where Z(p) = (2π i)pZ. Depending on the context,




      Ω

      X


      ∗




      {\displaystyle \Omega _{X}^{*}}

      is either the complex of smooth (i.e., C∞) differential forms or of holomorphic forms, respectively.
      The Deligne cohomology H qD,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










      Z







      ↓





      Ω

      X


      ∙
      ≥
      p




      →



      Ω

      X


      ∙








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



      Properties


      Deligne cohomology groups H qD (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





      Hdg


      p


      (
      X
      )
      ⊂

      H

      p
      ,
      p


      (
      X
      )


      {\displaystyle {\text{Hdg}}^{p}(X)\subset H^{p,p}(X)}

      of integral cohomology classes in




      H

      2
      p


      (
      X
      )


      {\displaystyle 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
      →

      J

      2
      p
      −
      1


      (
      X
      )
      →

      H


      D



      2
      p


      (
      X
      ,

      Z

      (
      p
      )
      )
      →


      Hdg


      2
      p


      (
      X
      )
      →
      0


      {\displaystyle 0\to J^{2p-1}(X)\to H_{\mathcal {D}}^{2p}(X,\mathbb {Z} (p))\to {\text{Hdg}}^{2p}(X)\to 0}




      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



      E


      {\displaystyle E}

      where




      π

      i


      (
      E
      )
      ⊗

      C

      =
      0


      {\displaystyle \pi _{i}(E)\otimes \mathbb {C} =0}

      for



      i


      {\displaystyle 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



      "Deligne-Beilinson cohomology" (PDF). Archived from the original (PDF) on 2020-06-03.
      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) [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, MR 2362847
      Esnault, Hélène; Viehweg, Eckart (1988), "Deligne-Beĭlinson cohomology" (PDF), Beĭlinson's conjectures on special values of L-functions, Perspect. Math., vol. 4, Boston, MA: Academic Press, pp. 43–91, ISBN 978-0-12-581120-0, MR 0944991
      Gajer, Pawel (1997), "Geometry of Deligne cohomology", Inventiones Mathematicae, 127 (1): 155–207, arXiv:alg-geom/9601025, Bibcode:1996InMat.127..155G, doi:10.1007/s002220050118, ISSN 0020-9910, S2CID 18446635
      Gomi, Kiyonori (2009), "Projective unitary representations of smooth Deligne cohomology groups", Journal of Geometry and Physics, 59 (9): 1339–1356, arXiv:math/0510187, Bibcode:2009JGP....59.1339G, doi:10.1016/j.geomphys.2009.06.012, ISSN 0393-0440, MR 2541824, S2CID 17437631

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