Definition 5.20.158FCD7BEEDD56

Definition 5.20.15.8FCD7BEEDD56 Assume that $L$ is a pseudo-effective line bundle on $X$. The set $\widetilde{\mathrm{FS}}(L)$ of generalized Fubini–Study metrics is the smallest subset of $\mathrm{PSH}(L)$ containing $\mathrm{FS}(L)$ which is closed under the following two operations:

1. B253418A2EBA

   $\mathbb {Q}$-convex combinations: if $h_1,h_2\in \widetilde{\mathrm{FS}}(L)$ and $t\in (0,1)$, then

   \[ h_1^ t \otimes h_2^{1-t}\in \widetilde{\mathrm{FS}}(L); \]
2. CCB3DD349505

   minima: if $h_1,h_2\in \widetilde{\mathrm{FS}}(L)$, then

   \[ \min \{ h_1,h_2\} \in \widetilde{\mathrm{FS}}(L). \]