Pome5929CFA4B87FLemma 5.37.5.5929CFA4B87F Let $q$ be quasi-plurisubharmonic on a complex manifold and suppose that $\nu (q,x)=0$ at every point. Then $\mathrm e^{-tq}$ is locally integrable for every $t>0$. For every quasi-plurisubharmonic $v$ and every $t>0$,
Proof. Fix a point $x$ and add a smooth correction to make $q$ plurisubharmonic on a bounded pseudoconvex coordinate neighborhood. Form the weighted Bergman space of proposition 5.35.1 at weight $t$. If every member vanished at $x$, the Cauchy estimates, together with the local upper bound for $q$, would give
near $x$. Its Bergman potential would consequently have Lelong number at least $1/t$, contradicting (5.35.3). There is therefore a weighted square-integrable holomorphic function nonzero at $x$. On a smaller neighborhood it is bounded away from zero, so its weighted norm proves integrability of $\mathrm e^{-tq}$. Smooth corrections do not change this conclusion. Thus $q\sim _{\mathcal I}0$; the addition rule in lemma 5.37.4 gives $v+q\sim _{\mathcal I}v$, which is precisely the asserted equality at every weight. $\square$
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