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Proposition 20.3.71C33DAF9C7870

Proposition 20.3.71.C33DAF9C7870 For every real $s>0$,

\[ \mathcal I(su_ X)=\mathcal O_ X(-\lfloor sb\rfloor D_ X), \qquad \mathcal I(su_ X)\mathcal O_ Y =\mathcal O_ Y(-\lfloor sb\rfloor D). \]

For every prime divisor $E$ on every smooth compact modification $\pi :Y'\to Y$, the intrinsic trace data are

\[ \tau _{Y,u_ X}(E)=b\operatorname {ord}_ E(\pi ^*D). \]

Nevertheless the ordinary type of $bG_ p+U_ L$ has no analytic representative.

Proof. Near $D_ X$, choose a transverse coordinate $z_1$. A germ whose exact transverse order is $j$ has a nonzero leading tangential coefficient. With $t=-\log |z_1|^2$, its radial integrability against $\mathrm e^{-su_ X}$ is determined by

\[ \exp \bigl (-(j+1-sb)t+s t^\alpha \bigr ). \]

Bounded errors in this exponent do not change convergence. It is integrable exactly when $j+1>sb$, and diverges at equality. Local holomorphic bounds give sufficiency, and a region where the leading tangential coefficient is nonzero gives necessity by Fubini. This proves the ambient ideal formula in the book's convention $|f|^2\mathrm e^{-su_ X}$. Restriction and pullback of this principal ideal give order $\lfloor sb\rfloor \operatorname {ord}_ E(\pi ^*D)$. Divide by $s$ and use theorem 20.3.2.

The residual $U_ L$ has trivial multiplier ideals at every weight, but is unbounded. At a generic point of $D$, an analytic type with divisorial coefficient $b$ is $b\log |z_1|^2+O(1)$. Our residual tends to $-\infty $ like $-t^\alpha $, excluding ordinary equivalence. This also covers $b=0$. $\square$

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