Let $X$ be a complex manifold, let $U \subset X$ be an [open set](/page/Open%20Set), let $L \to X$ be a holomorphic line bundle, and let $h$ be a smooth Hermitian metric on $L$. Let
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\begin{align*}
e: U \to L|_U
\end{align*}
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be a nowhere-vanishing holomorphic local frame. Suppose that there exists a smooth real-valued function
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\begin{align*}
\varphi: U \to \mathbb{R}
\end{align*}
for every $x \in U$. Let $\partial$ and $\bar{\partial}$ denote the Dolbeault operators on smooth forms on $U$. Let $\nabla^L$ denote the Chern connection of $(L,h)$, characterized by $(\nabla^L)^{0,1}=\bar{\partial}_L$ and metric compatibility with $h$. Let $A_e \in \Omega^{1,0}(U)$ be the local connection form defined by
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\begin{align*}
\nabla^L e=A_e\otimes e,
\end{align*}
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and let the local curvature form in the frame $e$ be