Example 20.3.54.D8423B168573 Set
\[ U([z_0:\cdots :z_ n]) =\log \frac{\sum _{i=0}^{n-2}|z_ i|^2}{\sum _{i=0}^ n|z_ i|^2}, \qquad P=[0:\cdots :0:1]\in B. \]
Then $\nu (U,B)=0$ and, for every $t>0$,
\[ \mathfrak b_ t=\mathfrak m_ P^{\max \{ \lfloor t-d+1\rfloor ,0\} }, \qquad \tau _{Y,u}(D)=\operatorname {ord}_ D(\mathfrak m_ P\mathcal O_ Z) \]
for every divisor $D$ on every smooth compact modification $Z\to Y$, with the ideal extended through $q$. The potential $q^*(U|_ B)$ is an exact realizer, singular along the elliptic fiber above $P$. For this explicit potential it suffices to take $a>1$, and
\[ \int _ Y\mathrm{MA}_{\Omega _{a,b}}(q^*(U|_ B))=2nb(a^ d-1). \]
It need not equal the normalized trace envelope.
Proof.
The singular locus of $U$ is the projective line defined by $z_0=\cdots =z_{n-2}=0$, of codimension $d$. Its smooth-center multiplier ideals are given by lemma 5.37.1; restriction to $B$ gives the displayed powers of $\mathfrak m_ P$. Dividing their orders by $t$ proves the trace formula. Exactness of the ordinary restriction follows from proposition 20.3.6, since the ambient pullback has analytic singularities.
On the point blowup $\sigma :\widetilde B\to B$, let $F$ be the exceptional divisor, $H=[\sigma ^*\kappa ]$, and let $\eta $ be the pullback of the Fubini–Study form by the resolved projection to $\mathbb P^{d-1}$. The logarithm defining $U|_ B$ gives
\[ \sigma ^*(a\kappa )+\mathrm{dd}^{\mathrm{c}}(U|_ B\circ \sigma ) =[F]+(a-1)\sigma ^*\kappa +\eta . \]
The residual form is smooth semipositive for $a>1$, with class $aH-F$. Its smooth top product off $F$ computes the non-pluripolar mass, and the smooth measure does not charge $F$. Since $H|_ F=0$, all mixed intersections involving both $H$ and $F$ vanish; $H^ d=1$ and $F^ d=(-1)^{d-1}$ give $(aH-F)^ d=a^ d-1$. Formula (20.3.53) now supplies the elliptic factor $2nb$.
$\square$
Pome
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