PomeCorollary 10.25.5.A12B874B6590 In the preceding setting, let $\theta $ be a smooth real $(1,1)$-form on $X$, and let $u\in \mathrm{PSH}(X,\theta )$ be smooth on $M$, with $T=\theta +\mathrm{dd}^{\mathrm{c}}u>0$ there. Suppose that, for some $\lambda >0$ and some positive $f\in C^\infty (M)\cap L^1(X,\omega ^ n)$,
\[ T^ n=\mathrm{e}^{\lambda u}f\omega ^ n\quad \hbox{on }M. \]
Assume that, near $D$, there are constants $c,C>0$, $q>1$, $a\geq 0$ and $b\in \mathbb R$ such that
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\begin{equation} f\geq cR^{-2}\tau ^{-q},\qquad u\geq -\frac{a}{\lambda }\log \tau -C,\qquad \det _\omega B_\tau \leq C\tau ^ b. \end{equation}
If $q+a+b\leq 2$, then $(M,g_ T)$ is complete.
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