[proofplan] We prove the assertion in local unitary frames. In such a frame, a unitary connection is represented by an imaginary-valued local $1$-form, so $\frac{i}{2\pi}F_\nabla$ is a globally defined real closed $2$-form. The identification of its de Rham class with the image of the integral first Chern class then follows from the rank-one case of the Chern-Weil integrality theorem. [/proofplan]

[step:Represent the unitary connection in local unitary frames]Let $(U_\alpha)_{\alpha\in A}$ be an [open cover](/page/Open%20Cover) of $M$ over which $L$ admits smooth unitary local frames. For each $\alpha\in A$, let \begin{align*} e_\alpha:U_\alpha\to L|_{U_\alpha} \end{align*} denote a nowhere-vanishing smooth section with Hermitian norm $1$. Define the local connection form \begin{align*} A_\alpha\in\Omega^1(U_\alpha;\mathbb C) \end{align*} by the identity \begin{align*} \nabla(f e_\alpha)=(d f+A_\alpha f)e_\alpha \end{align*} for every smooth function $f:U_\alpha\to\mathbb C$. Since $\nabla$ is unitary and $e_\alpha$ has constant Hermitian norm $1$, compatibility with the Hermitian metric gives \begin{align*} 0=d\langle e_\alpha,e_\alpha\rangle=\langle \nabla e_\alpha,e_\alpha\rangle+\langle e_\alpha,\nabla e_\alpha\rangle=A_\alpha+\overline{A_\alpha}. \end{align*} Thus $A_\alpha$ is imaginary-valued, so $A_\alpha\in\Omega^1(U_\alpha;i\mathbb R)$.[/step]

[guided]The first point is to translate the connection into scalar local data. Choose an open cover $(U_\alpha)_{\alpha\in A}$ of $M$ such that $L|_{U_\alpha}$ has a unitary smooth frame \begin{align*} e_\alpha:U_\alpha\to L|_{U_\alpha}. \end{align*} Because $L$ is a complex line bundle, every smooth local section over $U_\alpha$ has the form $f e_\alpha$ for a unique smooth function $f:U_\alpha\to\mathbb C$. The connection is therefore determined by what it does to $e_\alpha$. Define \begin{align*} A_\alpha\in\Omega^1(U_\alpha;\mathbb C) \end{align*} by \begin{align*} \nabla(f e_\alpha)=(d f+A_\alpha f)e_\alpha. \end{align*} This is just the Leibniz rule for a connection written in the frame $e_\alpha$. Now we use the word unitary. Since $e_\alpha$ has Hermitian norm $1$, the function $\langle e_\alpha,e_\alpha\rangle:U_\alpha\to\mathbb R$ is constant equal to $1$. Metric compatibility of $\nabla$ gives \begin{align*} d\langle e_\alpha,e_\alpha\rangle=\langle \nabla e_\alpha,e_\alpha\rangle+\langle e_\alpha,\nabla e_\alpha\rangle. \end{align*} Because $\nabla e_\alpha=A_\alpha e_\alpha$ and $\langle e_\alpha,e_\alpha\rangle=1$, this becomes \begin{align*} 0=A_\alpha+\overline{A_\alpha}. \end{align*} Hence every value of $A_\alpha$ lies in $i\mathbb R$. This is the local reason the normalized curvature form will be real after multiplication by $i$.[/guided]

[step:Compute the curvature and prove the normalized form is real] On $U_\alpha$, the curvature of $\nabla$ is represented in the frame $e_\alpha$ by \begin{align*} F_\nabla|_{U_\alpha}=dA_\alpha+A_\alpha\wedge A_\alpha. \end{align*} Since $A_\alpha$ is a scalar-valued $1$-form, $A_\alpha\wedge A_\alpha=0$. Hence \begin{align*} F_\nabla|_{U_\alpha}=dA_\alpha. \end{align*} Because $A_\alpha$ is imaginary-valued, $dA_\alpha$ is imaginary-valued. Therefore $\frac{i}{2\pi}F_\nabla|_{U_\alpha}$ is real-valued. Since this holds on every $U_\alpha$, the form \begin{align*} c_1(L,\nabla):=\frac{i}{2\pi}F_\nabla \end{align*} lies in $\Omega^2(M;\mathbb R)$. [/step]

[step:Check that the local curvature forms glue across frame changes] For $\alpha,\beta\in A$, define the unitary transition function \begin{align*} g_{\alpha\beta}:U_\alpha\cap U_\beta\to U(1) \end{align*} by \begin{align*} e_\beta=e_\alpha g_{\alpha\beta}. \end{align*} On $U_\alpha\cap U_\beta$, comparing the two formulas for $\nabla(f e_\beta)$ gives \begin{align*} A_\beta=A_\alpha+g_{\alpha\beta}^{-1}d g_{\alpha\beta}. \end{align*} Since $U(1)$ is abelian, the scalar $1$-form $g_{\alpha\beta}^{-1}d g_{\alpha\beta}$ satisfies \begin{align*} d(g_{\alpha\beta}^{-1}d g_{\alpha\beta})=0. \end{align*} Thus \begin{align*} dA_\beta=dA_\alpha \end{align*} on $U_\alpha\cap U_\beta$. Hence the local forms $dA_\alpha$ agree on overlaps and represent the global curvature form $F_\nabla$. [/step]

[step:Prove closedness from the local abelian curvature formula] For each $\alpha\in A$, the restriction of $F_\nabla$ to $U_\alpha$ is $dA_\alpha$. Therefore \begin{align*} d(F_\nabla|_{U_\alpha})=d(dA_\alpha)=0. \end{align*} Since exterior differentiation is local, $dF_\nabla=0$ on $M$. Multiplication by the constant $\frac{i}{2\pi}$ gives \begin{align*} d c_1(L,\nabla)=d\left(\frac{i}{2\pi}F_\nabla\right)=\frac{i}{2\pi}dF_\nabla=0. \end{align*} Thus $c_1(L,\nabla)$ is a closed real $2$-form. [/step]

[step:Identify the de Rham class with the integral first Chern class]We now use the Čech-de Rham comparison for the exponential sequence appearing in the theorem statement. Here $C^\infty_M(\mathbb R)$ denotes the sheaf of smooth real-valued functions on $M$, and $C^\infty_M(U(1))$ denotes the sheaf of smooth $U(1)$-valued functions on $M$: \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*} The unitary transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\to U(1)$ define a $C^\infty_M(U(1))$-valued Čech $1$-cocycle. After replacing the cover by a refinement on which logarithms exist, choose smooth maps \begin{align*} h_{\alpha\beta}:U_\alpha\cap U_\beta\to\mathbb R \end{align*} such that \begin{align*} g_{\alpha\beta}=\exp(2\pi i h_{\alpha\beta}). \end{align*} On triple overlaps define the integer-valued Čech $2$-cochain \begin{align*} n_{\alpha\beta\gamma}:=h_{\beta\gamma}-h_{\alpha\gamma}+h_{\alpha\beta}:U_\alpha\cap U_\beta\cap U_\gamma\to\mathbb Z. \end{align*} The equality $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$ implies that $(n_{\alpha\beta\gamma})$ is a locally constant integer-valued Čech $2$-cocycle, and by the stated convention its cohomology class is $c_1(L)$. Differentiating $g_{\alpha\beta}=\exp(2\pi i h_{\alpha\beta})$ gives \begin{align*} g_{\alpha\beta}^{-1}d g_{\alpha\beta}=2\pi i\,d h_{\alpha\beta}. \end{align*} The transformation law for connection forms gives \begin{align*} A_\beta-A_\alpha=2\pi i\,d h_{\alpha\beta}. \end{align*} Define real local $1$-forms \begin{align*} a_\alpha:=\frac{i}{2\pi}A_\alpha\in\Omega^1(U_\alpha;\mathbb R). \end{align*} Then \begin{align*} a_\beta-a_\alpha=-d h_{\alpha\beta}. \end{align*} Also define the global real $2$-form \begin{align*} \omega:=\frac{i}{2\pi}F_\nabla\in\Omega^2(M;\mathbb R). \end{align*} Since $F_\nabla|_{U_\alpha}=dA_\alpha$, one has \begin{align*} \omega|_{U_\alpha}=d a_\alpha. \end{align*} We use the following explicit sign convention for the Čech-de Rham total complex of the cover $(U_\alpha)_{\alpha\in A}$. For a Čech $p$-cochain $\eta$ with values in differential forms, the total differential is \begin{align*} D\eta=\delta\eta+(-1)^p d\eta, \end{align*} where the Čech coboundary is \begin{align*} (\delta b)_{\alpha\beta}=b_\beta-b_\alpha \end{align*} for a $0$-cochain $b=(b_\alpha)$, and \begin{align*} (\delta h)_{\alpha\beta\gamma}=h_{\beta\gamma}-h_{\alpha\gamma}+h_{\alpha\beta} \end{align*} for a $1$-cochain $h=(h_{\alpha\beta})$. With this convention, the displayed definition gives \begin{align*} \delta h=n. \end{align*} The equation $a_\beta-a_\alpha=-d h_{\alpha\beta}$ is exactly \begin{align*} \delta a=-d h, \end{align*} or equivalently $\delta a+d h=0$, which is the total-cocycle equation in bidegree $(1,1)$. Finally $d a_\alpha=\omega|_{U_\alpha}$ on each $U_\alpha$, and the already proved equality $d\omega=0$ gives the final total-cocycle equation. Thus $(n_{\alpha\beta\gamma},h_{\alpha\beta},a_\alpha,\omega)$ is a Čech-de Rham representative whose integral Čech component is $n$ and whose global de Rham component is $\omega$. By the Čech-de Rham comparison map with this total differential convention, the integral class represented by $(n_{\alpha\beta\gamma})$ maps to the de Rham class $[\omega]$. Therefore \begin{align*} [c_1(L,\nabla)]_{\mathrm{dR}}=\left[\frac{i}{2\pi}F_\nabla\right] \end{align*} is exactly the image of $c_1(L)\in H^2(M;\mathbb Z)$ under the canonical map to $H^2_{\mathrm{dR}}(M)$.[/step]

[guided]The last point is the comparison between two ways of encoding the same line bundle. The first way is topological. The sheaf $C^\infty_M(\mathbb R)$ assigns to each [open set](/page/Open%20Set) $V\subset M$ the smooth real-valued functions $V\to\mathbb R$, and the sheaf $C^\infty_M(U(1))$ assigns the smooth maps $V\to U(1)$. The exponential sequence \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*} defines the connecting homomorphism used in the theorem statement. The unitary transition functions \begin{align*} g_{\alpha\beta}:U_\alpha\cap U_\beta\to U(1) \end{align*} form a Čech $1$-cocycle. After refining the cover so that logarithms exist on all double overlaps, choose smooth maps \begin{align*} h_{\alpha\beta}:U_\alpha\cap U_\beta\to\mathbb R \end{align*} with \begin{align*} g_{\alpha\beta}=\exp(2\pi i h_{\alpha\beta}). \end{align*} On triple overlaps, the cocycle identity $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma}$ gives \begin{align*} \exp\bigl(2\pi i(h_{\beta\gamma}-h_{\alpha\gamma}+h_{\alpha\beta})\bigr)=1. \end{align*} Therefore \begin{align*} n_{\alpha\beta\gamma}:=h_{\beta\gamma}-h_{\alpha\gamma}+h_{\alpha\beta}:U_\alpha\cap U_\beta\cap U_\gamma\to\mathbb Z \end{align*} is a locally constant integer-valued Čech $2$-cocycle. By the convention in the statement, the class of $(n_{\alpha\beta\gamma})$ is $c_1(L)\in H^2(M;\mathbb Z)$. The second way is differential-geometric: the same bundle is equipped with a unitary connection, whose local connection forms are \begin{align*} A_\alpha\in\Omega^1(U_\alpha;i\mathbb R). \end{align*} On overlaps, the frame relation $e_\beta=e_\alpha g_{\alpha\beta}$ gives \begin{align*} A_\beta=A_\alpha+g_{\alpha\beta}^{-1}d g_{\alpha\beta}. \end{align*} Differentiating $g_{\alpha\beta}=\exp(2\pi i h_{\alpha\beta})$ gives \begin{align*} g_{\alpha\beta}^{-1}d g_{\alpha\beta}=2\pi i\,d h_{\alpha\beta}. \end{align*} Therefore the connection forms satisfy \begin{align*} A_\beta-A_\alpha=2\pi i\,d h_{\alpha\beta}. \end{align*} Define real local $1$-forms \begin{align*} a_\alpha:=\frac{i}{2\pi}A_\alpha\in\Omega^1(U_\alpha;\mathbb R). \end{align*} Multiplying the preceding transformation law by $\frac{i}{2\pi}$ gives \begin{align*} a_\beta-a_\alpha=-d h_{\alpha\beta}. \end{align*} Equivalently, \begin{align*} d h_{\alpha\beta}=a_\alpha-a_\beta. \end{align*} Since \begin{align*} F_\nabla|_{U_\alpha}=dA_\alpha, \end{align*} the real $2$-form \begin{align*} \omega:=\frac{i}{2\pi}F_\nabla\in\Omega^2(M;\mathbb R) \end{align*} satisfies \begin{align*} \omega|_{U_\alpha}=d a_\alpha. \end{align*} Now we spell out the sign convention instead of hiding it inside the phrase Čech-de Rham comparison. For a Čech $p$-cochain $\eta$ with values in differential forms, use the total differential \begin{align*} D\eta=\delta\eta+(-1)^p d\eta. \end{align*} For a $0$-cochain $b=(b_\alpha)$, this means \begin{align*} (\delta b)_{\alpha\beta}=b_\beta-b_\alpha, \end{align*} and for a $1$-cochain $h=(h_{\alpha\beta})$, it means \begin{align*} (\delta h)_{\alpha\beta\gamma}=h_{\beta\gamma}-h_{\alpha\gamma}+h_{\alpha\beta}. \end{align*} Therefore the definition of $n_{\alpha\beta\gamma}$ says exactly \begin{align*} \delta h=n. \end{align*} The sign-sensitive overlap equation is the next one. Since \begin{align*} a_\beta-a_\alpha=-d h_{\alpha\beta}, \end{align*} we have \begin{align*} (\delta a)_{\alpha\beta}=-d h_{\alpha\beta}. \end{align*} Equivalently, \begin{align*} \delta a+d h=0. \end{align*} This is precisely the total-cocycle equation in bidegree $(1,1)$, because $h$ has Čech degree $1$ and hence its de Rham differential enters the total differential with the sign $(-1)^1=-1$ when the equation is written as the descent relation from $n$ to $a$. Finally, \begin{align*} d a_\alpha=\omega|_{U_\alpha} \end{align*} on every $U_\alpha$, and $d\omega=0$ was proved in the preceding step. Thus the data $(n_{\alpha\beta\gamma},h_{\alpha\beta},a_\alpha,\omega)$ form the Čech-de Rham representative of the same total cohomology class: $n$ is the integral Čech cocycle, $h$ is the real logarithm cochain with $\delta h=n$, $a$ is the local real connection potential satisfying $\delta a=-d h$, and $\omega$ is the global curvature form satisfying $d a_\alpha=\omega|_{U_\alpha}$. Consequently the Čech-de Rham comparison sends the class represented by $(n_{\alpha\beta\gamma})$ to $[\omega]$. Hence the de Rham cohomology class of $\frac{i}{2\pi}F_\nabla$ is the image of $c_1(L)\in H^2(M;\mathbb Z)$ under the canonical map from integral cohomology to real de Rham cohomology.[/guided]

[step:Conclude the theorem] The preceding steps show that $c_1(L,\nabla)=\frac{i}{2\pi}F_\nabla$ is a real closed $2$-form and that its de Rham cohomology class is the canonical real de Rham image of $c_1(L)\in H^2(M;\mathbb Z)$. This is the asserted first Chern form formula for the Hermitian line bundle $L\to M$ with unitary connection $\nabla$. [/step]