Definition 5.20.3293669D813B8

Definition 5.20.3.293669D813B8 A Hermitian metric $h$ on $L$ is a family of Hermitian forms $(h_ x)_{x\in X}$, such that

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   for each $x\in X$, $h_ x$ is a Hermitian form on $L_ x$, and

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   for each local holomorphic section $s$ of $\mathcal{O}_ X(L)$, the map $x\mapsto |s(x)|^2_{h_ x}$ is smooth.

The pair $(L,h)$ is called a Hermitian line bundle. We shall write $\mathrm{dd}^{\mathrm{c}}h=c_1(L,h)$² for the first Chern form of $h$³, normalized so that

The map $x\mapsto |s(x)|_{h_ x}$ will be denoted by $|s|_ h$.

To be more precise, if $U\subseteq X$ is an open subset on which $L$ admits a nowhere vanishing holomorphic section $s$, then we define