Pome8065AC6F9F0FThe mass of a basic potential is computed by the closed base and the area of the elliptic fiber. The formula below holds for all $a,b>0$. It computes one representative; the target still fails the bounded mass property by corollary 23.1.36.
Proposition 20.3.52.13817C91E7B4 For every $v\in \mathrm{PSH}(B,a\kappa )$,
B5F41C0BB745
20.3.53
In particular, the trace envelope in (20.3.49) has mass between $2nb(a/2)^ d$ and $2nba^ d$.
Proof. Normalize $v\leq 0$ and put $v_ j=\max (v,-j)$. For bounded potentials, the horizontal current has rank at most $d=n-1$ and $\beta ^2=0$, so the bounded local product gives
This identity follows by local decreasing regularization on base charts and the bounded product convergence of section 13.14; it does not require $\beta $ to be closed. Restrict to $q^{-1}\{ v>-j\} $ and pass to the increasing non-pluripolar measures. Fiber integration gives (20.3.53), and (20.3.47) gives the two bounds. $\square$
Example 20.3.54.D8423B168573 Set
Then $\nu (U,B)=0$ and, for every $t>0$,
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
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
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 $2nb$. $\square$
Corollary 20.3.55.313F5BBD3487 The ideal singularity class of $\mathcal T_{\Omega _{a,b}}(u;Y)$ is independent of $a\geq 2C$ and $b>0$, and depends only on the multiplier-ideal class of $U$. These classes respect addition and positive scaling of the ambient data: for $U_1,U_2$ with zero generic Lelong number along $B$ and $\lambda >0$, in sufficiently large invariant backgrounds,
where the backgrounds on the individual envelopes are omitted from the notation.
Proof. All sufficiently large base backgrounds give the same divisorial Lelong numbers (20.3.47). On the compact Kähler base, equality of these numbers on all compact models is equivalent to equality of the multiplier ideals at every real positive weight; see the closed criterion recalled in 25.15. Hence all $w_ a$ have the same ideal class. The submersion identity of lemma 5.37.7 transfers the equality to $q^*w_ a$, which does not depend on $b$. Equal ambient multiplier ideals also give equal trace data.
By theorem 20.3.4, the trace data of $U_1+U_2$ is the sum of those of $U_1$ and $U_2$. The sum of two exact base realizers has precisely these divisorial Lelong numbers, so the same closed criterion identifies its ideal class with that of the base trace envelope for $U_1+U_2$. Positive scaling is identical. Pullback gives both assertions on $Y$. $\square$
The ideal-class conclusion here concerns the invariant backgrounds $\Omega _{a,b}$. Exactness in every larger Hermitian background follows from theorem 20.3.48, but does not by itself identify an ordinary or ideal class in all those backgrounds. The criterion in the proof is applied on $B$, not assumed on a general non-Kähler manifold.
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