Example 18.15.5.CB61780C4135 In every complex dimension $n\geq 3$, there is a projective manifold with the bounded mass property that admits a smooth positive nonclosed background $\Theta $ and normalized ordinary models $a\leq b$ such that
\[ \begin{gathered} P_\Theta [a]=a,\qquad P_\Theta [a]_{\mathcal I}=P_\Theta [b]=P_\Theta [b]_{\mathcal I}=b,\\ a\sim _{\mathcal I}b,\qquad a\not\sim b,\\ \underline{\operatorname {vol}}_\Theta (a) =\overline{\operatorname {vol}}_\Theta (a) =\underline{\operatorname {vol}}_\Theta (b) =\overline{\operatorname {vol}}_\Theta (b)=0. \end{gathered} \]
In particular, equality of the upper volumes of a type and of its ideal model does not force equality of the envelopes when the ideal model has zero lower volume.
Proof.
Let $Y=\mathbb P^1\times \mathbb P^1$, with projections $\pi _1,\pi _2$, and let $\sigma $ be the probability Fubini–Study form on $\mathbb P^1$. Choose a positive nonconstant smooth function $f$ on the first factor and put
\[ \beta =\pi _1^*\sigma +(f\circ \pi _1)\pi _2^*\sigma . \]
This form is positive, and $\mathrm d\beta =\pi _1^*(\mathrm df)\wedge \pi _2^*\sigma \ne 0$. Let $w_0$ be the normalized diffuse polar model of example 13.28.14, and put $w=w_0\circ \pi _1$. Then $w\in \mathrm{PSH}(Y,\beta )$ and
\[ P_\beta [w]=w,\qquad P_\beta [w]_{\mathcal I}=0. \]
Indeed, replace an ordinary competitor $v\leq 0$ by $\max (v,w)$. On every horizontal section $y\mapsto (y,z)$, where $\beta $ restricts to $\sigma $, this maximum is a $\sigma $-plurisubharmonic function, not identically $-\infty $ since it dominates $w_0$, and it is at least as singular as $w_0$. Its restriction is therefore at most $P_\sigma [w_0]=w_0$. This proves $v\leq w$ on $Y$; the reverse inequality for the envelope holds because $w$ is itself a competitor. All multiplier ideals of $w_0$ are trivial, so lemma 5.37.7 shows that the same holds for $w$. Since $\beta $ is positive, zero is a competitor for the ideal envelope of $w$, which proves the second identity.
Set $k=n-2$ and $M=Y\times \mathbb P^ k$, with projections $p,r$. On the second factor take the probability Fubini–Study form $\kappa $, a hyperplane $D$, and its normalized potential $\psi $ with $\kappa +\mathrm{dd}^{\mathrm{c}}\psi =[D]$. Put
\[ \Theta =p^*\beta +r^*\kappa ,\qquad \Psi =\psi \circ r,\qquad a=\Psi +w\circ p,\qquad b=\Psi . \]
The form $\Theta $ is positive and not closed. A sufficiently large multiple of the closed positive product form dominates $\Theta $, so the bounded mass property follows from Stokes' theorem for that closed form and the metric independence in proposition 13.14.15.
Apply proposition 18.14.1 to $p:M\to Y$ and the divisor $Y\times D$. The induced zero obstacle is zero because $\sup \psi =0$. The two identities on the base give all four identities upstairs, and both functions have supremum zero. Ideal projection preserves the ideal type, so $a\sim _{\mathcal I}b$. Moreover $a\leq b$, while their difference restricts to the unbounded function $w$ on any section outside the divisor, so $a\not\sim b$. Since $\dim Y=2<n$, the same proposition gives zero extended mass for every ordinary representative of $a$ or $b$, which proves the volume assertions.
$\square$
Pome
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