Pome87F63F4EB505Example 18.15.3.87F63F4EB505 Take $C=\mathbb P^1$ and $Z=\mathbb P^ k$, $k\geq 1$, with their standard Fubini–Study forms, and let $D$ be a hyperplane. In this product setting there is a potential $u$ without an analytic ordinary type such that
and the two envelopes of $u$ agree below every smooth obstacle. Their common model for the zero obstacle has zero singularity volume.
Proof. Use the unbounded curve potential $w$ constructed in the proof of proposition 18.10.1, whose two envelopes are zero. A logarithmic potential of a hyperplane on $Z$ gives $\psi $ with $\kappa +\mathrm{dd}^{\mathrm{c}}\psi =[D]$ and supremum zero. Set $u=\Psi +w\circ p$ and apply proposition 18.15.1, corollary 18.14.5. Fixing $z\notin D$ shows that $u-\Psi =w\circ p$ is unbounded, so $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. $\square$
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