Deligne cohomology – or Deligne-Beilinson cohomology is an abelian sheaf cohomology that models ordinary differential cohomology.
The standard Deligne complex (of abelian sheaves) is under the Dold-Kan correspondence the sheaf of n-groupoids of smooth n-functors from the path n-groupoid to the $n$-fold delooping $\mathbf{B}^n U(1)$:
Smooth Deligne cohomology in degree $n$, of a smooth space $X$ is cohomology with coefficients in $\bar \mathbf{B}^n U(1)$.
Here the notation on the right is as at the end of motivation for sheaves, cohomology and higher stacks.
This is a realization of the differential refinement (or smooth extension) $\bar H^n(X,\mathbb{Z})$ of the integral cohomology $H^n(X, \mathbb{Z})$ of $X$ in terms of abelian sheaf cohomology.
Recall that analogous to how $H^n(X,\mathbb{Z})$ classifies line $(n-1)$-bundles and equivalently line $(n-2)$-gerbes on $X$, $\bar H^n(X, \mathbb{Z})$ classifies line $(n-2)$-gerbes with connection.
Accordingly, the Deligne complex of sheaves $\mathbb{Z}(n)^\infty_D$ is a complex of sheaves of differential forms.
For $k \in \mathbb{N}$ write $\Omega^k(-) : U \mapsto \Omega^k(U)$ for the [[sheaf] of smooth differential $k$-forms on $X$ and $C^\infty(-,V)$ for the sheaf of smooth $V$-valued functions on $X$.
The degree $(n+1)$ Deligne complex is the complex of sheaves
Often it is useful to consider the quasi-isomorphic complex
Here $C^\infty(-,U(1)) \stackrel{d log}{\to} \Omega^1(-)$ is the morphism of sheaves induced by regarding a $U(1) \simeq \mathbb{R}/\mathbb{Z}$-valued function locally as a $\mathbb{R}$-valued function and applying the deRham differential $d$ to that.
The obvious morphism of complexes
clearly induces isomorphism on homology groups: the homology in degree $n$ is locally constant $\mathbb{R}$-valued functions modulo locally constant $\mathbb{Z}$-valued functions in the first case and constant $U(1)$-valued functions in the second case, which is the same.
Deligne cohomology in degree $n+1$ of $X$ is the cohomology (which is abelian sheaf cohomology in this case) with coefficients in $\bar \mathbf{B}^n U(1)$.
Here the notation on the right is motivated from the discussion at the end of motivation for sheaves, cohomology and higher stacks.
There are two natural morphisms of abelian cohomology groups out of Deligne cohomology:
These are induced from the canonical morphisms of coefficient objects
given by
and
given by
These two morphisms exhibit Deligne cohomology as a refinement in differential cohomology of ordinary (i.e. integral Eilenberg-MacLane) cohomology, in that the diagram
is the cohomology of a homotopy pullback diagram, i.e. satisfies the axioms described at differential cohomology.
There is a natural way to understand the Deligne complex of sheaves as a sheaf which assigns to each patch the Lie $n$-groupoid of smooth higher parallel transport n-functors. This perspective is helpful for understanding how Deligne cohomology relates to the bigger picture of differential cohomology.
We start by discussing this in low degree.
There is path groupoid $P_1(X)$ whose smooth space of objects is $X$ and whose smooth space of morphisms is a space of classes of smooth paths in $X$. Every smooth 1-form $A \in \Omega^1(X)$ induces a smooth functor $tra_A : P_1(X) \to \mathbf{B}U(1)$ from $P_1(X)$ to to the smooth groupoid $\mathbf{B} U(1)$ with one object and $U(1)$ as its smooth space of morphisms by sending each path $\gamma : [0,1] \to X$ to $\exp (2 \pi i\int_0^1 \gamma^* A)$. This map from 1-forms to smooth functors turns out to be bijective: every smooth functor of this form uniquely arises this way. Similarly, one finds that smooth natural transformation $\eta_f : tra_A \to tra_{A'}$ between two such functors is in components precisely a smooth function $f : X \to U(1)$ such that $A' = A + d log f$.
Since the analogous statements are true for every open subset $U \subset X$ this defines a sheaf of Lie groupoids
By the Dold-Kan correspondence this sheaf of groupoids corresponds to a sheaf of complexes of groups. This complex of sheaves is nothing but the degree 2 Deligne complex
This way Deligne cohomology is realized as computing the stackification of the pre-stack $Funct^\infty(P_1(-), \mathbf{B}(1))$ of smooth $U(1)$-valued parallel transport functors.
The identification generalizes: for all $n$ there is a path n-groupoid $P_n(X)$ whose $k$-morphisms are $k$-dimensional smooth paths in $X$. Smooth $n$-functors $tra_C : _n(X) \to \mathbf{B}^n U(1)$ are canonically identified with smooth $n$-forms $C \in \Omega^n(X)$ and under the Dold-Kan correspondence the Deligne-complex in degree $n+1$ is identified with the sheaf of $n$-groupoids of such smooth $n$-functors
See
The full proof for $n=1$ this is in
for $n=2$ in
For more on this see infinity-Chern-Weil theory introduction.
For higher $n$ there is as yet no detailed proof in the literature, but the low dimensional proofs have obvious generalizations.
See Beilinson-Deligne cup-product.
moduli spaces of line n-bundles with connection on $n$-dimensional $X$
The Deligne complex is naturally defined in smooth differential geometry as well as in complex analytic geometry as well as in algebraic geometry over the complex numbers. In the spirit of GAGA it is of interest to know how Deligne cohomology in these different settings relates.
One useful statement is: given an smooth algebraic variety over the complex numbers, then a sufficient condition for a complex-analytic Deligne cocycle over its analytification to lift to an algebraic Deligne cocycle is that its curvature form is an algebraic form (Esnault 89, corollary 1.3).
As described in some detail at electromagnetic field in abelian higher gauge theories the background field naturally arises as a Čech–Deligne cocycle, i.e. a Čech cocycle representative with values in the Deligne complex.
Degree 2 Deligne cohomology classifies $U(1)$-principal bundles with connection. The Deligne complex $\bar \mathbf{B}U(1)$ in this case coincides with the groupoid of Lie-algebra valued forms for the Lie algebra of $U(1)$.
Degree 3 Deligne cohomology classifies bundle gerbes with connection.
Degree 4 Deligne cohomology classifies bundle 2-gerbes with connection. In particular Chern-Simons bundle 2-gerbes whose degree 4 curvature characteristic class is a multiple of the Pontryagin 4-form on some $SO(n)$-principal bundle.
Deligne cohomology was introduced in complex analytic geometry (by a chain complex of holomorphic differential forms) in
with applications to Hodge theory and intermediate Jacobians. The same definition appears in
Barry Mazur, William Messing, Universal extensions and one-dimensional crystalline cohomology, Springer lecture notes 370, 1974
Michael Artin, Barry Mazur, section III.1 of Formal Groups Arising from Algebraic Varieties, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 numdam, MR56:15663
under the name “multiplicative de Rham complex” (and in the context of studying its deformation theory by Artin-Mazur formal groups). The theory was further developed in
with the application to Beilinson regulators. Later the evident version of the Deligne complex in differential geometry over smooth manifolds gained more attention and is still referred to as “Deligne cohomology”.
Surveys and introductions in the context of differential geometry include
Jean-Luc Brylinski, section 5 of Loop Spaces, Characteristic Classes and geometric Quantization, Birkhaeuser
Ulrich Bunke, section 3 of Differential cohomology (arXiv:1208.3961)
Review with more emphasis on complex analytic geometry and the theory of (Beilinson 85) with more details spelled out is in
Hélène Esnault, Eckart Viehweg, Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)
Hélène Esnault, On the Loday-symbol in the Deligne-Beilinson cohomology, K-theory 3, 1-28, 1989 (pdf)
See also
Chris Peters, Jozef Steenbrink, section 7.2 of Mixed Hodge Structures, Ergebisse der Mathematik (2008) (pdf)
Claire Voisin, section 12 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3
See also the references given at differential cohomology hexagon – Deligne coefficients.