[proofplan] We first compare the local expression in two holomorphic frames on the same [open set](/page/Open%20Set). The transition function is a nowhere-vanishing [holomorphic function](/page/Holomorphic%20Function), and locally its logarithm splits $\log |g|^2$ into a holomorphic term plus an anti-holomorphic term, so applying $\partial\bar\partial$ kills it. This proves that the local forms glue. We then check directly that the glued form is real and closed. Finally, for two metrics we write their ratio as $e^{-\psi}$ and compute that the corresponding forms differ by an exact $d$-form, which proves metric independence of the de Rham class. [/proofplan] [step:Compare the metric functions under a change of holomorphic frame] Let $U \subset X$ be an open set, and let $e: U \to L|_U$ and $\tilde e: U \to L|_U$ be two nowhere-vanishing holomorphic frames. Since both frames are nowhere vanishing and span the same holomorphic line bundle, there is a unique nowhere-vanishing holomorphic function \begin{align*} g: U &\to \mathbb{C}^* \\ x &\mapsto g(x) \end{align*} such that $\tilde e = e g$. Define the metric functions \begin{align*} h_e: U &\to \mathbb{R}_{>0}, & x &\mapsto h_x(e(x),e(x)), \\ h_{\tilde e}: U &\to \mathbb{R}_{>0}, & x &\mapsto h_x(\tilde e(x),\tilde e(x)). \end{align*} By Hermitian sesquilinearity of $h_x$ on the one-dimensional complex [vector space](/page/Vector%20Space) $L_x$, for every $x \in U$, \begin{align*} h_{\tilde e}(x) &= h_x(e(x)g(x),e(x)g(x)) \\ &= |g(x)|^2 h_x(e(x),e(x)) \\ &= |g(x)|^2 h_e(x). \end{align*} Thus \begin{align*} \log h_{\tilde e}=\log h_e+\log |g|^2. \end{align*} [/step] [step:Show that the transition term has zero $\partial\bar\partial$] Fix a point $p \in U$. Since $g: U \to \mathbb{C}^*$ is holomorphic and nowhere zero, there is an open neighbourhood $V \subset U$ of $p$ and a holomorphic function \begin{align*} \ell: V &\to \mathbb{C} \end{align*} such that $g|_V=\exp \ell$. On $V$, \begin{align*} \log |g|^2 = \log(\exp \ell \cdot \overline{\exp \ell}) = \ell+\overline{\ell}. \end{align*} Because $\ell$ is holomorphic, $\bar\partial \ell=0$. Because $\overline{\ell}$ is anti-holomorphic, $\partial\overline{\ell}=0$. Therefore \begin{align*} \partial\bar\partial \log |g|^2 &= \partial\bar\partial(\ell+\overline{\ell}) \\ &= \partial(\bar\partial \ell)+\partial(\bar\partial\overline{\ell}) \\ &= 0+\partial\bar\partial\overline{\ell} \\ &= 0, \end{align*} where the last equality follows from $\partial\overline{\ell}=0$ and the identity $\partial\bar\partial=-\bar\partial\partial$ on smooth forms. Since $p \in U$ was arbitrary, $\partial\bar\partial \log |g|^2=0$ on all of $U$. [/step] [step:Glue the local forms to a global form] Using the identity from the preceding step, on $U$ we obtain \begin{align*} -\frac{i}{2\pi}\partial\bar\partial\log h_{\tilde e} &= -\frac{i}{2\pi}\partial\bar\partial(\log h_e+\log |g|^2) \\ &= -\frac{i}{2\pi}\partial\bar\partial\log h_e. \end{align*} Thus the local forms $\Theta_{h,e}$ are independent of the choice of holomorphic frame. Let $(U_\alpha)_{\alpha \in A}$ be an open cover of $X$ by holomorphic trivialising neighbourhoods, and for each $\alpha \in A$ choose a nowhere-vanishing holomorphic frame \begin{align*} e_\alpha: U_\alpha \to L|_{U_\alpha}. \end{align*} The agreement just proved implies that \begin{align*} \Theta_{h,e_\alpha}|_{U_\alpha \cap U_\beta} = \Theta_{h,e_\beta}|_{U_\alpha \cap U_\beta} \end{align*} for all $\alpha,\beta \in A$. Hence there is a unique global smooth $2$-form \begin{align*} \Theta_h \in \Omega^2(X;\mathbb{C}) \end{align*} whose restriction to $U_\alpha$ is $\Theta_{h,e_\alpha}$ for every $\alpha \in A$. Since each local representative has type $(1,1)$, the global form $\Theta_h$ has type $(1,1)$. [/step] [step:Verify that the global form is real and closed] Let $U \subset X$ be a holomorphic trivialising open set with nowhere-vanishing holomorphic frame $e: U \to L|_U$. Define \begin{align*} \varphi: U &\to \mathbb{R} \\ x &\mapsto \log h_e(x). \end{align*} Since $h_e$ is positive and smooth, $\varphi$ is a smooth real-valued function. First, $\Theta_h$ is real. Complex conjugation interchanges $\partial$ and $\bar\partial$ on forms and fixes $\varphi$, so \begin{align*} \overline{\partial\bar\partial \varphi} = \bar\partial\partial \varphi = -\partial\bar\partial \varphi. \end{align*} Therefore \begin{align*} \overline{-\frac{i}{2\pi}\partial\bar\partial\varphi} = \frac{i}{2\pi}\overline{\partial\bar\partial\varphi} = \frac{i}{2\pi}(-\partial\bar\partial\varphi) = -\frac{i}{2\pi}\partial\bar\partial\varphi. \end{align*} Thus $\Theta_h|_U$ is real, and since this is local, $\Theta_h \in \Omega^2(X;\mathbb{R})$. Second, $\Theta_h$ is closed. On $U$, \begin{align*} d\Theta_h &= -\frac{i}{2\pi}(\partial+\bar\partial)\partial\bar\partial\varphi \\ &= -\frac{i}{2\pi}\left(\partial^2\bar\partial\varphi+\bar\partial\partial\bar\partial\varphi\right) \\ &= -\frac{i}{2\pi}\left(0-\partial\bar\partial^2\varphi\right) \\ &= 0, \end{align*} using $\partial^2=0$, $\bar\partial^2=0$, and $\bar\partial\partial=-\partial\bar\partial$. Since closedness is local, $d\Theta_h=0$ on $X$. [/step] [step:Compare the forms attached to two Hermitian metrics] Let $h_0$ and $h_1$ be two smooth Hermitian metrics on $L$. Define the smooth positive function \begin{align*} \rho: X &\to \mathbb{R}_{>0} \end{align*} by the condition \begin{align*} h_1(v,w)=\rho(x)h_0(v,w) \end{align*} for every $x \in X$ and every $v,w \in L_x$. This is well-defined because each fibre $L_x$ is one-dimensional and both Hermitian forms are positive definite. Define the smooth real-valued function \begin{align*} \psi: X &\to \mathbb{R} \\ x &\mapsto -\log \rho(x). \end{align*} Then $h_1=e^{-\psi}h_0$. Let $U \subset X$ be a holomorphic trivialising open set with nowhere-vanishing holomorphic frame $e: U \to L|_U$. Let \begin{align*} (h_j)_e: U &\to \mathbb{R}_{>0}, & x &\mapsto (h_j)_x(e(x),e(x)) \end{align*} for $j \in \{0,1\}$. From $h_1=e^{-\psi}h_0$ we have \begin{align*} \log (h_1)_e=\log (h_0)_e-\psi|_U. \end{align*} Therefore \begin{align*} \Theta_{h_1}|_U-\Theta_{h_0}|_U &= -\frac{i}{2\pi}\partial\bar\partial\log (h_1)_e +\frac{i}{2\pi}\partial\bar\partial\log (h_0)_e \\ &= \frac{i}{2\pi}\partial\bar\partial(\psi|_U). \end{align*} Because this identity is local and the right-hand side is the restriction of the global form $\frac{i}{2\pi}\partial\bar\partial\psi$, we obtain \begin{align*} \Theta_{h_1}-\Theta_{h_0} = \frac{i}{2\pi}\partial\bar\partial\psi. \end{align*} [/step] [step:Rewrite the metric change as an exact real de Rham form] Define the real $1$-form \begin{align*} \alpha := -\frac{i}{4\pi}(\partial\psi-\bar\partial\psi) \in \Omega^1(X;\mathbb{R}). \end{align*} The form $\alpha$ is real because $\psi$ is real-valued and complex conjugation interchanges $\partial\psi$ and $\bar\partial\psi$. Using $d=\partial+\bar\partial$, $\partial^2=0$, $\bar\partial^2=0$, and $\bar\partial\partial=-\partial\bar\partial$, we compute \begin{align*} d\alpha &= -\frac{i}{4\pi}(\partial+\bar\partial)(\partial\psi-\bar\partial\psi) \\ &= -\frac{i}{4\pi}\left(\partial^2\psi-\partial\bar\partial\psi+\bar\partial\partial\psi-\bar\partial^2\psi\right) \\ &= -\frac{i}{4\pi}\left(0-\partial\bar\partial\psi-\partial\bar\partial\psi-0\right) \\ &= \frac{i}{2\pi}\partial\bar\partial\psi. \end{align*} Combining this with the previous step gives \begin{align*} \Theta_{h_1}-\Theta_{h_0}=d\alpha. \end{align*} Thus $\Theta_{h_1}$ and $\Theta_{h_0}$ determine the same class in $H^2_{\mathrm{dR}}(X;\mathbb{R})$. [/step] [step:Construct the Chern connection and compute its curvature locally] Choose a holomorphic trivialising cover $(U_\alpha)_{\alpha \in A}$ with holomorphic frames $e_\alpha: U_\alpha \to L|_{U_\alpha}$. On overlaps $U_\alpha\cap U_\beta$, define the holomorphic transition functions \begin{align*} g_{\alpha\beta}: U_\alpha\cap U_\beta &\to \mathbb{C}^* \end{align*} by $e_\beta=e_\alpha g_{\alpha\beta}$. A connection $\nabla^h$ on $L$ is compatible with the holomorphic structure if its $(0,1)$-part is the Dolbeault operator $\bar\partial_L$, and it is compatible with the Hermitian metric $h$ if \begin{align*} d\bigl(h(s,t)\bigr)=h(\nabla^h s,t)+h(s,\nabla^h t) \end{align*} for all local smooth sections $s,t$ of $L$. The [Chern connection](/page/Chern%20Connection) theorem applies because $L \to X$ is a holomorphic line bundle and $h$ is a smooth Hermitian metric; it gives a unique connection $\nabla^h$ satisfying these two compatibility conditions. In the frame $e_\alpha$, write \begin{align*} \nabla^h e_\alpha = A_\alpha \otimes e_\alpha \end{align*} for the local connection $1$-form $A_\alpha \in \Omega^1(U_\alpha;\mathbb{C})$. The condition $(\nabla^h)^{0,1}=\bar\partial_L$ and the holomorphicity of $e_\alpha$ imply that the $(0,1)$-part of $A_\alpha$ is zero, so $A_\alpha \in \Omega^{1,0}(U_\alpha)$. Applying metric compatibility to $s=t=e_\alpha$ gives \begin{align*} dh_{e_\alpha} &= h(A_\alpha e_\alpha,e_\alpha)+h(e_\alpha,A_\alpha e_\alpha) \\ &= (A_\alpha+\overline{A_\alpha})h_{e_\alpha}. \end{align*} Taking the $(1,0)$-part and using that $A_\alpha$ has type $(1,0)$ gives \begin{align*} \partial h_{e_\alpha}=A_\alpha h_{e_\alpha}, \end{align*} and since $h_{e_\alpha}>0$, \begin{align*} A_\alpha=\partial\log h_{e_\alpha}. \end{align*} Thus the curvature $F_{\nabla^h}\in \Omega^2(X;\mathbb{C})$ restricts on $U_\alpha$ to \begin{align*} F_{\nabla^h}|_{U_\alpha} &= dA_\alpha \\ &= (\partial+\bar\partial)\partial\log h_{e_\alpha} \\ &= \bar\partial\partial\log h_{e_\alpha} \\ &= -\partial\bar\partial\log h_{e_\alpha}, \end{align*} using $\partial^2=0$ and $\bar\partial\partial=-\partial\bar\partial$. Hence \begin{align*} \frac{i}{2\pi}F_{\nabla^h}|_{U_\alpha} = -\frac{i}{2\pi}\partial\bar\partial\log h_{e_\alpha} = \Theta_h|_{U_\alpha}. \end{align*} Since the restrictions agree on every member of the cover, the global normalized curvature form satisfies \begin{align*} \frac{i}{2\pi}F_{\nabla^h}=\Theta_h. \end{align*} [/step] [step:Apply the Chern-Weil representative theorem for complex line bundles] The [Chern-Weil representative theorem for the first Chern class](/page/Chern-Weil%20Theory) states that if $L \to X$ is a smooth complex line bundle and $\nabla$ is a connection on $L$, then the closed real de Rham form $\frac{i}{2\pi}F_\nabla$ represents the image of the integral first Chern class under the coefficient map \begin{align*} H^2(X;\mathbb{Z}) \to H^2(X;\mathbb{R}). \end{align*} Equivalently, under the [Čech-de Rham comparison](/page/Cech-de%20Rham%20Comparison), this class is the same class obtained from the transition functions $g_{\alpha\beta}: U_\alpha\cap U_\beta \to \mathbb{C}^*$ by taking local logarithms and forming the standard integer Čech $2$-cocycle defining $c_1(L)$. We verify the hypotheses of this representative theorem. The holomorphic line bundle $L \to X$ is in particular a smooth complex line bundle over the smooth manifold underlying $X$. The preceding step constructs the Chern connection $\nabla^h$ on $L$, and the earlier closedness calculation proves that its normalized curvature form $\Theta_h=\frac{i}{2\pi}F_{\nabla^h}$ is a closed real $2$-form. Therefore the Chern-Weil representative theorem applies to the connection $\nabla^h$ and yields \begin{align*} [\Theta_h] = \left[\frac{i}{2\pi}F_{\nabla^h}\right] = c_1(L) \in H^2(X;\mathbb{R}). \end{align*} Combined with the metric-independence proved above, this identifies the common de Rham cohomology class of the local forms with the real first Chern class. This completes the proof. [/step]