Definition 6.14.9D022961325A2

Definition 6.14.9.D022961325A2 A holomorphic line bundle $L$ on $X$ is given by an open cover $(V_ i)$ of $X$ and local frames $e_ i$ over $V_ i$ with transition functions $e_ i=g_{ij}e_ j$ on $V_ i\cap V_ j$, where $g_{ij}$ is holomorphic and nowhere zero and $g_{ij}g_{jk}=g_{ik}$ on triple overlaps. A smooth quadratic Hermitian metric on $L$ assigns to each frame a smooth positive squared norm $|e_ i|^2$ on $V_ i$, such that $|e_ i|^2=|g_{ij}|^2|e_ j|^2$ on $V_ i\cap V_ j$. Its curvature is the closed real $(1,1)$-form equal to $-\mathrm{dd}^{\mathrm{c}}\log |e_ i|^2$ on $V_ i$.