Let $M$ be a smooth manifold, let $L\to M$ be a smooth Hermitian complex line bundle, and let $\nabla$ be a unitary connection on $L$. Let $F_\nabla\in\Omega^2(M;i\mathbb R)$ denote the scalar curvature $2$-form determined in a local unitary frame $e$ by $\nabla e=Ae$ and $F_\nabla=dA$. Define the integral first Chern class $c_1(L)\in H^2(M;\mathbb Z)$ by the connecting homomorphism for the exponential sequence of sheaves

\begin{align*} 0\longrightarrow \mathbb Z\longrightarrow C^\infty_M(\mathbb R)\xrightarrow{\exp(2\pi i\,\cdot)} C^\infty_M(U(1))\longrightarrow 0, \end{align*}

where $C^\infty_M(\mathbb R)$ is the sheaf of smooth real-valued functions on $M$ and $C^\infty_M(U(1))$ is the sheaf of smooth $U(1)$-valued functions on $M$. Then

\begin{align*} c_1(L,\nabla):=\frac{i}{2\pi}F_\nabla \end{align*}

is a closed real differential $2$-form on $M$. Its de Rham cohomology class is the image of $c_1(L)$ under the canonical coefficient and de Rham comparison map

\begin{align*} H^2(M;\mathbb Z)\longrightarrow H^2(M;\mathbb R)\longrightarrow H^2_{\mathrm{dR}}(M). \end{align*}