Proposition 16.28.72AA248AFFA815

Proof. Use the smooth nonpositive $\omega $-plurisubharmonic functions $u_ j$, $j\geq 1$, of theorem 13.14.18 and their convergent series $U=\sum _{j\geq 1}2^{-j}u_ j$. For each integer $N\geq 1$, write

The head $A_ N$ is smooth, and since the finite partial sums of the tail decrease, $V_ N\in \mathrm{PSH}(X,\delta _ N\omega )$; the tail is not identically $-\infty $, because $U=A_ N+V_ N$ is not.

Fix $b>1$ and choose $N$ so large that $b\delta _ N\leq 1$. Then

is $\omega $-plurisubharmonic, because $\mathrm{dd}^{\mathrm{c}}h_ b\geq -b\delta _ N\omega \geq -\omega $. Moreover, $h_ b\leq bU$. Thus $h_ b\in \mathcal A_ b(U,0)$ for every $b$, and since $U\leq 0$, this proves $U\in \mathcal E_{\mathrm{env}}(X,\omega ,0)$. For $\varepsilon U$, use the tails $\varepsilon V_ N$, and choose $N$ with $b\varepsilon \delta _ N\leq 1$. By proposition 16.28.65, the singularity envelope of each of these potentials is $P_\omega [0]=0$. Their infinite mass and trivial multiplier ideals are given by corollary 13.14.19. $\square$