PomeCorollary 18.1.9.4662D2032EC0 Let $\theta $ be a closed real smooth $(1,1)$-form on $X$ representing a big cohomology class. Let $(\varphi _ j)_{j\in I}$ be a net of $\mathcal{I}$-good potentials in $\mathrm{PSH}(X,\theta )$ such that $\varphi _ j\xrightarrow {d_ S}\varphi $. Then $\varphi $ is $\mathcal{I}$-good.
Corollary 18.1.94662D2032EC0
Proof. By adding a Kähler form to $\theta $, and using 18.1.8, corollary 16.9.36, we may assume that $\varphi _ j,\varphi \in \mathrm{PSH}(X,\theta )_{>0}$ for all $j\in I$. It follows from theorem 18.1.2 that
\[ \int _ X \theta _{\varphi _ j}^ n=\operatorname{vol}\theta _{\varphi _ j} \]
for all $j\in I$. Taking limit with respect to $j$ with the help of theorem 16.9.57, we conclude that
\[ \int _ X \theta _{\varphi }^ n=\operatorname{vol}\theta _{\varphi }. \]
Therefore, by theorem 18.1.2 again, we find that $\varphi $ is $\mathcal{I}$-good. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.