[proofplan] We compute the form in local holomorphic frames and first prove that the local expressions agree on overlaps. Closedness and type are then immediate from the identities for $\partial$ and $\bar{\partial}$, while reality follows by conjugating $\partial\bar{\partial}\varphi_i$ for real $\varphi_i$. Finally, on a good cover we compare the curvature representative with the usual integer Čech cocycle of the line bundle by an explicit Čech-de Rham zig-zag. [/proofplan] [step:Show that the local curvature forms agree on overlaps] Choose a good holomorphic trivializing cover $(U_i)_{i\in I}$ of $M$. For each $i\in I$, let \begin{align*} e_i:U_i&\to L|_{U_i} \end{align*} be a nowhere-vanishing holomorphic frame, and define $\varphi_i:U_i\to\mathbb{R}$ by $|e_i|_h^2=e^{-\varphi_i}$. On $U_i\cap U_j$, define the holomorphic transition function \begin{align*} g_{ij}:U_i\cap U_j&\to \mathbb{C}^{\times} \end{align*} by $e_i=g_{ij}e_j$. Then \begin{align*} e^{-\varphi_i}=|e_i|_h^2=|g_{ij}|^2|e_j|_h^2=|g_{ij}|^2e^{-\varphi_j}, \end{align*} so \begin{align*} \varphi_j-\varphi_i=\log |g_{ij}|^2. \end{align*} Since $g_{ij}$ is nowhere zero and holomorphic, locally $\log |g_{ij}|^2=\log g_{ij}+\log \overline{g_{ij}}$, where $\log g_{ij}$ is holomorphic and $\log\overline{g_{ij}}$ is anti-holomorphic. Hence \begin{align*} \partial\bar{\partial}\log |g_{ij}|^2=0. \end{align*} Therefore \begin{align*} \partial\bar{\partial}\varphi_j=\partial\bar{\partial}\varphi_i \end{align*} on $U_i\cap U_j$. The local forms \begin{align*} \omega_i:=\frac{i}{2\pi}\partial\bar{\partial}\varphi_i \end{align*} therefore glue to a global form $c_1(L,h)\in\Omega^2(M;\mathbb{C})$. [guided] The first point is that the definition cannot depend on the chosen holomorphic frame. Let $e_i:U_i\to L|_{U_i}$ and $e_j:U_j\to L|_{U_j}$ be two nowhere-vanishing holomorphic frames. On the overlap $U_i\cap U_j$, the transition function is the holomorphic map \begin{align*} g_{ij}:U_i\cap U_j&\to \mathbb{C}^{\times} \end{align*} defined by $e_i=g_{ij}e_j$. The metric functions $\varphi_i$ and $\varphi_j$ are defined by \begin{align*} |e_i|_h^2=e^{-\varphi_i}, \qquad |e_j|_h^2=e^{-\varphi_j}. \end{align*} Using $e_i=g_{ij}e_j$ and the Hermitian property of $h$, we get \begin{align*} e^{-\varphi_i} = |e_i|_h^2 = |g_{ij}e_j|_h^2 = |g_{ij}|^2|e_j|_h^2 = |g_{ij}|^2e^{-\varphi_j}. \end{align*} Taking logarithms gives \begin{align*} \varphi_j-\varphi_i=\log |g_{ij}|^2. \end{align*} Now we use the holomorphicity of $g_{ij}$. Because $g_{ij}$ never vanishes, locally there is a holomorphic logarithm $\log g_{ij}$. On such a smaller [open set](/page/Open%20Set), \begin{align*} \log |g_{ij}|^2=\log g_{ij}+\log \overline{g_{ij}}. \end{align*} The function $\log g_{ij}$ is holomorphic, so $\bar{\partial}\log g_{ij}=0$. The function $\log\overline{g_{ij}}$ is anti-holomorphic, so $\partial\log\overline{g_{ij}}=0$. Hence \begin{align*} \partial\bar{\partial}\log |g_{ij}|^2=0. \end{align*} Applying $\partial\bar{\partial}$ to $\varphi_j-\varphi_i=\log |g_{ij}|^2$ gives \begin{align*} \partial\bar{\partial}\varphi_j-\partial\bar{\partial}\varphi_i=0. \end{align*} Thus the local forms \begin{align*} \omega_i:=\frac{i}{2\pi}\partial\bar{\partial}\varphi_i \end{align*} agree on every overlap and define a single global smooth $2$-form on $M$, denoted $c_1(L,h)$. [/guided] [/step] [step:Verify that the global form is closed, real, and of type $(1,1)$] Each local form $\omega_i=\frac{i}{2\pi}\partial\bar{\partial}\varphi_i$ is of type $(1,1)$ by construction. Since $d=\partial+\bar{\partial}$ and $\partial^2=\bar{\partial}^2=0$, \begin{align*} d\omega_i &= \frac{i}{2\pi}(\partial+\bar{\partial})\partial\bar{\partial}\varphi_i \\ &= \frac{i}{2\pi}\left(\partial^2\bar{\partial}\varphi_i+\bar{\partial}\partial\bar{\partial}\varphi_i\right) \\ &=0. \end{align*} Thus $dc_1(L,h)=0$. Because $\varphi_i$ is real-valued, \begin{align*} \overline{\partial\bar{\partial}\varphi_i} = \bar{\partial}\partial\varphi_i = -\partial\bar{\partial}\varphi_i. \end{align*} Therefore \begin{align*} \overline{\frac{i}{2\pi}\partial\bar{\partial}\varphi_i} = \frac{i}{2\pi}\partial\bar{\partial}\varphi_i, \end{align*} so $c_1(L,h)$ is real. [/step] [step:Relate the curvature form to the transition cocycle] On each $U_i$, define the local Chern connection one-form \begin{align*} \alpha_i:U_i&\to T^*U_i\otimes_{\mathbb{R}}\mathbb{C},\\ x&\mapsto -\partial\varphi_i(x). \end{align*} Its curvature is \begin{align*} F_i=d\alpha_i = -(\partial+\bar{\partial})\partial\varphi_i = -\bar{\partial}\partial\varphi_i = \partial\bar{\partial}\varphi_i. \end{align*} Hence \begin{align*} c_1(L,h)|_{U_i} = \frac{i}{2\pi}F_i = -\frac{1}{2\pi i}F_i. \end{align*} On $U_i\cap U_j$, using $\varphi_j-\varphi_i=\log |g_{ij}|^2$, we obtain \begin{align*} \alpha_j-\alpha_i &= -\partial\varphi_j+\partial\varphi_i\\ &= -\partial(\varphi_j-\varphi_i)\\ &= -\partial\log |g_{ij}|^2\\ &= -d\log g_{ij}. \end{align*} Since the cover is good, choose smooth functions \begin{align*} a_{ij}:U_i\cap U_j&\to\mathbb{C} \end{align*} such that \begin{align*} \exp(2\pi i a_{ij})=g_{ij}. \end{align*} Then $d\log g_{ij}=2\pi i\, da_{ij}$, so \begin{align*} \alpha_j-\alpha_i=-2\pi i\, da_{ij}. \end{align*} [/step] [step:Identify the Čech coboundary as the integral first Chern cocycle] On a triple overlap $U_i\cap U_j\cap U_k$, the transition functions satisfy \begin{align*} g_{ij}g_{jk}g_{ki}=1. \end{align*} Using $\exp(2\pi i a_{ij})=g_{ij}$, define \begin{align*} n_{ijk}:U_i\cap U_j\cap U_k&\to\mathbb{Z},\\ x&\mapsto a_{ij}(x)+a_{jk}(x)+a_{ki}(x). \end{align*} The image lies in $\mathbb{Z}$ because \begin{align*} \exp(2\pi i n_{ijk})=g_{ij}g_{jk}g_{ki}=1. \end{align*} Since each triple overlap is connected in a good cover and $n_{ijk}$ is continuous with values in the discrete set $\mathbb{Z}$, the function $n_{ijk}$ is constant. The integer Čech $2$-cocycle $(n_{ijk})$ represents $c_1(L)\in H^2(M;\mathbb{Z})$ by the standard exponential-sequence construction of the first Chern class. [/step] [step:Read the Čech-de Rham zig-zag to obtain the de Rham class] The preceding computations give the Čech-de Rham chain \begin{align*} F_i=d\alpha_i, \qquad \alpha_j-\alpha_i=-2\pi i\, da_{ij}, \qquad a_{ij}+a_{jk}+a_{ki}=n_{ijk}. \end{align*} In the Čech-de Rham double complex for the good cover $(U_i)$, this means that the global closed form \begin{align*} -\frac{1}{2\pi i}F_i = \frac{i}{2\pi}\partial\bar{\partial}\varphi_i \end{align*} corresponds under the Čech-de Rham comparison to the real Čech cocycle $(n_{ijk})$. Thus, by the Čech-de Rham comparison theorem (citing a result not yet in the wiki: Čech-de Rham comparison theorem), \begin{align*} [c_1(L,h)]_{\mathrm{dR}} = [(n_{ijk})]_{\mathbb{R}} = \rho(c_1(L)). \end{align*} This proves that $c_1(L,h)$ is a closed real $(1,1)$-form whose de Rham cohomology class is the image of the integral first Chern class. [/step]