[proofplan] We compute everything in the chosen holomorphic frame $e$. The Hermitian metric is represented by the positive scalar function $H=\exp(-\varphi)$. Metric compatibility and the condition $(\nabla^L)^{0,1}=\bar{\partial}_L$ determine the local Chern connection form as $A_e=H^{-1}\partial H$. Differentiating $H=\exp(-\varphi)$ gives $A_e=-\partial\varphi$, and applying $\bar{\partial}$ gives the curvature. The final equality follows from the Dolbeault anticommutation identity on functions. [/proofplan]

[step:Represent the metric by a positive scalar function in the holomorphic frame] Define the smooth function \begin{align*} H: U \to \mathbb{R}_{>0} \end{align*} by \begin{align*} H(x)=h_x(e(x),e(x)). \end{align*} By hypothesis, \begin{align*} H=\exp(-\varphi). \end{align*} Since $e$ is a holomorphic frame and $L$ has rank one, every local section of $L|_U$ can be written uniquely as $f e$ for a smooth function $f:U\to\mathbb{C}$. [/step]

[step:Apply the Chern connection formula in the holomorphic frame]Let $A_e\in \Omega^{1,0}(U)$ denote the local connection one-form determined by \begin{align*} \nabla^L e=A_e\otimes e. \end{align*} Because $e$ is holomorphic and $(\nabla^L)^{0,1}=\bar{\partial}_L$, the connection form has type $(1,0)$. To determine it, apply metric compatibility to the pair of sections $e$ and $e$: \begin{align*} dH=h(\nabla^L e,e)+h(e,\nabla^L e). \end{align*} Substituting $\nabla^L e=A_e\otimes e$ gives \begin{align*} dH=A_e H+\overline{A_e}H. \end{align*} Taking the $(1,0)$ component of this identity yields \begin{align*} \partial H=A_e H. \end{align*} Since $H:U\to\mathbb{R}_{>0}$ is everywhere positive, multiplication by $H^{-1}$ is defined, and therefore \begin{align*} A_e=H^{-1}\partial H. \end{align*} Using $H=\exp(-\varphi)$ and the chain rule for $\partial$, we get \begin{align*} \partial H=\partial(\exp(-\varphi))=-\exp(-\varphi)\partial\varphi. \end{align*} Multiplying by $H^{-1}=\exp(\varphi)$ gives \begin{align*} A_e=H^{-1}\partial H=\exp(\varphi)(-\exp(-\varphi)\partial\varphi)=-\partial\varphi. \end{align*} Therefore \begin{align*} \nabla^L e=-\partial\varphi\otimes e. \end{align*}[/step]

[guided]The point of using the holomorphic frame $e$ is that the Chern connection has a simple local formula in such a frame. We first name the connection form: define $A_e\in\Omega^{1,0}(U)$ by the identity \begin{align*} \nabla^L e=A_e\otimes e. \end{align*} The Chern connection is the unique connection compatible with both the holomorphic structure and the Hermitian metric. Since $e$ is a holomorphic frame and $(\nabla^L)^{0,1}=\bar{\partial}_L$, the local form $A_e$ has type $(1,0)$ rather than an arbitrary complex one-form. Now use metric compatibility. The metric coefficient is the smooth positive function $H:U\to\mathbb{R}_{>0}$ given by $H(x)=h_x(e(x),e(x))$. Metric compatibility applied to the two local sections $e$ and $e$ says \begin{align*} dH=h(\nabla^L e,e)+h(e,\nabla^L e). \end{align*} Using the defining identity $\nabla^L e=A_e\otimes e$, this becomes \begin{align*} dH=A_eH+\overline{A_e}H. \end{align*} Because $A_e$ has type $(1,0)$, the $(1,0)$ part of this equality is \begin{align*} \partial H=A_eH. \end{align*} The positivity of $H$ allows us to divide by $H$, so \begin{align*} A_e=H^{-1}\partial H. \end{align*} This is the local Chern connection formula in the holomorphic frame, and it has now been obtained directly from the two defining properties of the Chern connection. Here $H=\exp(-\varphi)$. Applying the chain rule to the smooth real-valued function $\varphi:U\to\mathbb{R}$ gives \begin{align*} \partial H=\partial(\exp(-\varphi))=-\exp(-\varphi)\partial\varphi. \end{align*} Since $H^{-1}=\exp(\varphi)$, substitution into the Chern connection formula yields \begin{align*} A_e=H^{-1}\partial H=\exp(\varphi)(-\exp(-\varphi)\partial\varphi)=-\partial\varphi. \end{align*} Returning to the defining identity for $A_e$, we conclude \begin{align*} \nabla^L e=A_e\otimes e=-\partial\varphi\otimes e. \end{align*}[/guided]

[step:Differentiate the connection form to obtain the curvature] By the stated curvature convention for the Chern connection in a holomorphic frame, \begin{align*} \Theta_L=\bar{\partial}A_e. \end{align*} Substituting $A_e=-\partial\varphi$ gives \begin{align*} \Theta_L=\bar{\partial}(-\partial\varphi)=-\bar{\partial}\partial\varphi. \end{align*} Thus the first curvature identity holds. [/step]

[step:Use the Dolbeault anticommutation identity on functions] The Dolbeault operators satisfy \begin{align*} \partial\bar{\partial}f+\bar{\partial}\partial f=0 \end{align*} for every smooth function $f:U\to\mathbb{C}$. Applying this identity to the smooth real-valued function $\varphi:U\to\mathbb{R}$ gives \begin{align*} -\bar{\partial}\partial\varphi=\partial\bar{\partial}\varphi. \end{align*} Combining this with the previous step gives \begin{align*} \Theta_L=-\bar{\partial}\partial\varphi=\partial\bar{\partial}\varphi. \end{align*} Together with $\nabla^L e=-\partial\varphi\otimes e$, this proves the claimed local formula. [/step]