Proposition 5.16.13 (Nadel)926B8027EE55

Proof. On each coordinate ball, the multiplier-ideal coherence theorem proves that the germs $f$ for which $|f|^2\exp(-\varphi)$ is locally integrable form a coherent analytic ideal sheaf. This is exactly the definition of $\mathcal I(\varphi)$ used here. These local sheaves agree on overlaps, since local integrability is unchanged by a holomorphic coordinate change with a nonvanishing Jacobian. Coherence is local, so $\mathcal I(\varphi)$ is coherent on $X$. In particular the argument uses the weight $\exp(-\varphi)$, not $\exp(-2\varphi)$.