[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 $A_\alpha\in\Omega^1(U_\alpha;i\mathbb R)$.
[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]
[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 $H^2_{\mathrm{dR}}(M)$.
[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]
[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]
