Pome055CAE14F80FProposition 16.4.1.055CAE14F80F Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$ to $X$. Then for any $\varphi \in \mathrm{PSH}(X,\theta )$, we have
In particular, a potential $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ is model if and only if $\pi ^*\varphi \in \mathrm{PSH}(Y,\pi ^*\theta )_{>0}$ is model.
Proof. By proposition 5.17.8, every $\chi \in \mathrm{PSH}(Y,\pi ^*\theta )$ is of the form $\chi =\pi ^*\psi $ for a unique $\psi \in \mathrm{PSH}(X,\theta )$. Since $\pi $ is surjective, $\chi \leq 0$ if and only if $\psi \leq 0$. Moreover,
Thus the candidates defining $P_{\pi ^*\theta }[\pi ^*\varphi ]$ are exactly the pull-backs of the candidates defining $P_\theta [\varphi ]$, and hence
Finally, the non-pluripolar mass is preserved under bimeromorphic pull-back by proposition 13.13.4. Therefore $\varphi $ is model of positive mass if and only if $\pi ^*\varphi $ is. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.