Example 18.14.868E397A3AF5C

Proof. Let $(Y,\omega )$ be the Hopf threefold of theorem 23.1.1. On $\mathbb P^1$, let $\sigma $ be the probability Fubini–Study form, choose a point $D$ and a logarithmic potential $\psi $ with $\sigma +\mathrm{dd}^{\mathrm{c}}\psi =[D]$ and $\sup \psi =0$. On $M=Y\times \mathbb P^1$, with projections $p,r$, put

Take the potential $U\in \mathcal E_{\mathrm{env}}(Y,\omega ,0)$ of infinite mass from proposition 16.28.72, which satisfies $P_\omega [U]=0$. Its multiplier ideals are trivial, so also $P_\omega [U]_{\mathcal I}=0$. Set $u=\Psi +U\circ p$. Here $\widehat0=0$, so proposition 18.14.1, corollary 18.14.5 give the envelope identities, the equality below all obstacles and the vanishing of the singularity volumes. Moreover $u\in \mathcal E_{\mathrm{env}}(M,\Theta ,\Psi )$ by proposition 18.14.4. Since $U$ has infinite extended mass, it is unbounded; restriction to a section outside $H$ therefore proves $u\not\sim \Psi $. If a potential $a$ with analytic singularities were ordinarily equivalent to $u$, proposition 18.4.8 would give $u\sim a\sim P_\Theta [a]=P_\Theta [u]=\Psi $, a contradiction.

Finally, let $V_ j$ be the smooth potentials of theorem 23.1.1. Their pullbacks are bounded and $\Theta $-plurisubharmonic, and proposition 20.3.59 gives

Thus $\Theta $ itself fails the bounded mass property. $\square$