PomeCorollary 10.28.10.15454CA8E897 Among all semipositive metrics on $L_0$ that extend, $h_*$ has the least singular weights: every such metric has weights $\psi $ satisfying $\psi \leq \psi _*+C$ in compatible local frames, where $\psi _*$ is the weight of $h_*$; and $h_*$ itself extends. The assertion concerns metrics that extend, not all semipositive metrics on $L_0$.
Corollary 10.28.1015454CA8E897
Proof. By theorem 10.28.9, such a metric has the form $h_\ell \otimes f^*h_ B$. Its weight difference from $h_*$ is the pullback of the global quasi-plurisubharmonic difference between the weights of $h_ B$ and $h_{\mathrm{FS}}$, which is bounded above on $\mathbb P^1$. This proves the inequality. That $h_*$ extends is lemma 10.28.7. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.