An ambient residual can be invisible to every multiplier ideal even when its restriction to $Y$ is identically $-\infty $. The restricted ideals then recover the imposed logarithmic divisor directly.
Proposition 20.3.8.C4231F8A37FD Let $q\in \mathrm{QPSH}(X)$ satisfy (18.27.1), and let $\Delta $ and $\sigma _\Delta $ be as in theorem 18.27.4. For every $t>0$,
\[ \mathcal I\bigl (t(\sigma _\Delta +q)\bigr ) =\mathcal I(t\sigma _\Delta ). \]
If $\Delta $ has simple normal crossings and $Y$ is contained in none of its positive-coefficient components, then for every divisor $E$ on every smooth connected compact modification $\rho :Z\to Y$,
\[ \tau _{Y,\sigma _\Delta +q}(E) =\sum _ ja_ j\operatorname {ord}_ E\rho ^*(D_ j|_ Y). \]
The formula imposes no condition on the naive restriction $q|_ Y$.
Proof.
The first equality follows from lemma 18.27.2, lemma 5.37.5. In normal crossing coordinates, the weight for $t\sigma _\Delta $ is a smooth positive factor times $\prod _ j|z_ j|^{-2ta_ j}$. One-variable polar integration gives the exact formula
\[ \mathcal I(t\sigma _\Delta ) =\mathcal O_ X\left(-\sum _ j\lfloor ta_ j\rfloor D_ j\right). \]
Indeed, integrability of a germ of transverse order $k$ requires $k+1>ta_ j$, so $k\geq \lfloor ta_ j\rfloor $; equality at the preceding order is divergent when $ta_ j$ is an integer. Necessity follows on an open set of the other variables where the leading coefficient is nonzero. After division by these powers, the remaining product weight is integrable, proving sufficiency.
Since $Y$ is contained in no $D_ j$, each restricted defining equation is a nonzero holomorphic germ on $Y$. Its pullback to $Z$ is nonzero and has a finite nonnegative order $N_{j,E}$ along $E$. Restriction of the displayed principal ideal and then pullback therefore give the exact order $\sum _ j\lfloor ta_ j\rfloor N_{j,E}$. Dividing by $t$ proves the trace formula, including along all real weights. Moreover $\nu (\sigma _\Delta +q,Y)=0$: outside the finitely many divisors both summands have zero point Lelong number. Thus these orders are the intrinsic data of theorem 20.3.2. When $\Delta =0$, the ideal is $\mathcal O_ X$ and every trace is zero.
$\square$
Pome
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