Definition 16.8.2033F50367BB68

Definition 16.8.20.33F50367BB68 Let $L$ be a pseudo-effective line bundle on $X$. An elementary metric on $L$ is a psh metric $h$ on $L$ such that there is a generalized Fubini–Study metric $h'$ on $L$ such that

The set of elementary metrics on $L$ is denoted by $\mathrm{Ele}(L)$.

We also say $\mathrm{dd}^{\mathrm{c}}h$ is elementary. If we have fixed a Hermitian metric $h_0$ on $L$, and if we represent $h$ as $h_0\exp (-\varphi )$, we also say the quasi-psh function $\varphi $ is elementary.