Lemma 5.43.19C07C7BB26EF8

Proof. By lemma 5.25.1, the functions $v_ j$ are uniformly bounded above on relatively compact subsets of $X$. The pullbacks are therefore locally uniformly bounded above and have the common smooth lower curvature bound $-\pi ^*\theta $. After a smooth local correction, the compactness theorem for plurisubharmonic functions shows that every subsequence has a further subsequence that either converges in $L^1_{\mathrm{loc}}$ or tends locally uniformly to $-\infty $ on each connected coordinate region in $Y$.

Every such region meets the complement of the exceptional locus. There $\pi $ is a biholomorphism, and change of variables gives $L^1_{\mathrm{loc}}$ convergence to $v\circ \pi $, which excludes convergence to $-\infty $. Each subsequential limit agrees with $v\circ \pi $ off the exceptional locus, hence almost everywhere, since this analytic set has smooth volume zero. Thus the limit is unique, and applying this argument on a countable cover by coordinate regions proves convergence of the whole sequence in $L^1_{\mathrm{loc}}(Y)$. $\square$