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,
\[ \mathcal T(u_1+u_2;Y)\sim _{\mathcal I} \mathcal T(u_1;Y)+\mathcal T(u_2;Y), \qquad \mathcal T(\lambda u_1;Y)\sim _{\mathcal I} \lambda \mathcal T(u_1;Y), \]
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 $Y$.
$\square$
Pome
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