Corollary 11.41.739C3764F36BD

Proof. If such a component existed, retain a sufficiently small positive multiple $E$ of it. Choose a canonical logarithm $\ell _ E$ and a smooth closed representative $\gamma _ E$ with $[E]=\gamma _ E+\mathrm{dd}^{\mathrm{c}}\ell _ E$. The multiple can be chosen so that $\theta -\gamma _ E>0$. Divisorial removal gives a canonical $(\theta -\gamma _ E)$-psh representative of $u-\ell _ E$, whose finite supremum implies $u\preceq \ell _ E$. The preceding theorem would then give

a contradiction. $\square$