Proposition 10.6.9E5F620306821

Proof. In a local frame $e$ write $s_ i=f_ ie^ m$. The local weight $-\log |e|_ h^2=\varphi -\log |e|_{h_0}^2$ of the metric equals $m^{-1}\max _ i(\log |f_ i|^2+\lambda _ i)$, which does not involve $h_0$; it is plurisubharmonic and not identically $-\infty $, since $X$ is connected and some $s_ i\ne 0$. It differs by a bounded function from $m^{-1}\log \sum _ i\mathrm e^{\lambda _ i}|f_ i|^2$, so it has analytic singularities with the rational coefficient $1/m$; the pole set is the common zero set of the $s_ i$.

Consider two such metrics, given by sections $s_ i$ of $L^ m$ with constants $\lambda _ i$ and sections $r_ j$ of $L^ l$ with constants $\mu _ j$. Replacing $s_ i$ by $s_ i^{M/m}$ and $\lambda _ i$ by $(M/m)\lambda _ i$, and similarly for the $r_ j$, where $M$ is a common multiple of $m$ and $l$, does not change the potentials, so we may assume that both are given in degree $M$. The maximum of the two potentials, which corresponds to the minimum of the metrics, is then given by the union of the two lists. For a rational $t=a/b\in (0,1)$, the combination $t\varphi +(1-t)\psi $ of the two maxima is the maximum over all pairs $(i,j)$ of the corresponding combinations, that is,

where the sections $s_ i^ ar_ j^{b-a}$ of $L^{Mb}$ are not all zero and the constants are rational. Thus $\mathrm{FS}(L)$ is closed under both operations, and by induction under every finite sequence of them, so the generated set is $\mathrm{FS}(L)$ itself. If no positive tensor power of $L$ has a nonzero section, no presentation as in definition 10.6.8 exists. $\square$