Smooth Deligne cohomology model

Fix a good open cover of a finite-dimensional Hausdorff second-countable smooth manifold $M$ without boundary. In degree $k\ge 1$, smooth Deligne cohomology is the degree-$k$ hypercohomology of the cohomological complex of sheaves $\mathbb{Z}\to {\mathrm{\Omega }}^0\to {\mathrm{\Omega }}^1\to \cdots \to {\mathrm{\Omega }}^{k-1}$, with $\mathbb{Z}$ in degree $0$ and ${\mathrm{\Omega }}^j$ in degree $j+1$. Equivalently it is degree-$(k-1)$ hypercohomology of the shifted complex of smooth circle-valued functions and real forms, $C^{\mathrm{\infty }}(-,U(1))\stackrel{d\log \mathrm{/}(2\pi i)}{\to }{\mathrm{\Omega }}^1\to \cdots \to {\mathrm{\Omega }}^{k-1}$, with $U(1)=\mathbb{R}\mathrm{/}\mathbb{Z}$ under $t\mapsto e^{2\pi it}$. For $k=1$ this circle-valued complex consists of $C^{\mathrm{\infty }}(-,U(1))$ alone in degree $0$, with zero differential; for $k\ge 2$, its degree-$j$ term is ${\mathrm{\Omega }}^j$ for $1\le j\le k-1$, and all other terms are zero.

De rham cochain complex

Ordered Čech cochain complex of a cover

Fixed-cover Čech cohomology

A good cover here means an ordered locally finite open cover $\mathcal{U}=(U_i)$ such that every nonempty finite intersection admits a smooth contraction to a point, together with a chosen smooth partition of unity subordinate to $\mathcal{U}$. The sheaf $\mathcal{U}(1)$ assigns to an open set $V$ the abelian group $C^{\mathrm{\infty }}(V,U(1))$, with restrictions by composition and pointwise multiplication. The first map in the integer-form complex is inclusion of locally constant integer functions into smooth real functions; the later maps are the exterior derivative.

Use cohomological grading. For a cochain map $g:C^{\bullet }\to D^{\bullet }$, set $\mathrm{Cone}(g{)}^n=C^{n+1}\oplus D^n$ with differential $(x,y)\mapsto (-d_{C}x,d_{D}y-gx)$, and set $K[-1{]}^n=K^{n-1}$ with differential $-d_K$. The stupid truncation ${\sigma }_{\ge k}{\mathrm{\Omega }}^{\bullet }$ has term ${\mathrm{\Omega }}^j$ for $j\ge k$ and zero for $j<k$; ${\sigma }_{<k}{\mathrm{\Omega }}^{\bullet }$ has term ${\mathrm{\Omega }}^j$ for $j<k$ and zero for $j\ge k$, with the differential out of degree $k-1$ set to zero. The equivalent cone model is ${\mathcal{D}}_k=\mathrm{Cone}(f)[-1]$, with $f:\mathbb{Z}\oplus {\sigma }_{\ge k}{\mathrm{\Omega }}^{\bullet }\to {\mathrm{\Omega }}^{\bullet }$ given by $f(z,\alpha )=z-\alpha$. Projection to $\mathrm{Cone}(\mathbb{Z}\to {\sigma }_{<k}{\mathrm{\Omega }}^{\bullet })[-1]$ is a quasi-isomorphism: its kernel is $\mathrm{Cone}(-{\mathrm{i}\mathrm{d}}_{{\sigma }_{\ge k}{\mathrm{\Omega }}^{\bullet }})[-1]$, which is contractible. The projected complex has $\mathbb{Z}$ in degree $0$ and ${\mathrm{\Omega }}^j$ in degree $j+1$ for $0\le j<k$; multiplying the ${\mathrm{\Omega }}^j$ term by $(-1{)}^j$ gives the integer-form differential with first map inclusion and later maps $d$. The Cech total complex has terms ${\mathrm{Tot}}^n={⨁}_{p+q=n}{C}^p(\mathcal{U},{\mathcal{D}}_k^q)$ and differential ${\delta }_{\mathrm{C}\mathrm{e}\mathrm{c}\mathrm{h}}+(-1{)}^p{d}_{\mathcal{D}}$ on bidegree $(p,q)$. Its degree-$k$ cohomology is the stated model. Refinement maps are induced by cover refinement; the resulting map on cohomology is independent of the refinement function. The alternative circle-valued complex is obtained from the exponential sequence of sheaves $0\to \mathbb{Z}\to {\mathrm{\Omega }}^0\to \mathcal{U}(1)\to 0$; for $k\ge 2$ its first differential is $d\log (g)=(2\pi i{)}^{-1}{g}^{-1}dg$, whereas for $k=1$ there is no form term or outgoing differential. The characteristic map $I$ and curvature map $R$ are the projections induced by the cone model; $a$ is the connecting map from the form component. These are model definitions; exactness under the explicit good-cover Cech-to-singular comparison input is stated separately in the Remarks.

This definition fixes the Deligne grading and normalization. Conditional exactness under the good-cover Cech-to-singular comparison input is treated in Exact sequence for smooth Deligne cohomology. Equivalence with the smooth-singular Cheeger–Simons character model is a separate theorem, Deligne and Cheeger–Simons models agree under geometric cycle inputs, whose local proof requires the explicitly named geometric input in this batch.