Definition 5.20.8A43467D63B4A

Definition 5.20.8.A43467D63B4A A (singular) plurisubharmonic metric (or psh metric for short)⁴ $h$ on $L$ is a family $(h_ x)_{x\in X}$ such that

1. DCF7D606ADB1

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

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   there is a Hermitian metric $h_0$ on $L$ and $\varphi \in \mathrm{PSH}(X,c_1(L,h_0))$ such that for each $x\in X$ and each $v\in L_ x$, we have

   6C8BC3AC251D 5.20.9

   \begin{equation} |v|^2_{h_ x}= \left\{ \begin{aligned} 0,& \quad \textup{if }v=0;\\ |v|^2_{h_{0,x}}\mathrm{e}^{-\varphi (x)},& \quad \textup{if }v\neq 0. \end{aligned} \right. \end{equation}

The (first) Chern current of $h$ is by definition

The set of psh metrics on $L$ will be denoted by $\mathrm{PSH}(L)$.