5.43.21 Multiplier ideals under modificationsF75CE9F0BB10

Proof. The assertion is local on $X$. In coordinates and local frames, the map sends a holomorphic coefficient $f$ to $(f\circ \pi )\operatorname {Jac}\pi $. By the change-of-variables formula, local integrability of

are equivalent. For the implication from integrability on $Y$ to integrability near a point of $X$, take a smaller relatively compact neighborhood of the point; by properness, its inverse image has compact closure, so finitely many charts on which the integrability holds suffice. Smooth positive coordinate densities are comparable on these charts, and analytic exceptional sets have smooth volume zero, so they do not change the integrals.

It remains to show that every section on $Y$ is of the form $J_\pi \pi ^*s$. Divide it by $J_\pi $ off $D$ and use the inverse of $\pi $ there; this gives a holomorphic section of $M$ off $S$. Since $S$ has codimension at least two, Hartogs' extension theorem gives a holomorphic section $s$ on $X$. Its image $J_\pi \pi ^*s$ agrees with the given section off $D$, hence everywhere by the identity theorem. The integrability equivalence then places $s$ in the required multiplier ideal. The same argument on each open set proves the sheaf identity as well as the global identification. $\square$

Proof. The reverse implication follows at every positive weight from the sheaf identity in lemma 5.43.22: tensor the inclusion of ideals on $Y$ with $R_\pi $ and apply the left-exact direct image functor. The proof of that sheaf identity is local on the target, since properness makes each fiber compact and gives the required finite cover by integrability charts; compactness of the manifolds themselves is not needed.

For the forward implication, fix a coordinate neighborhood $W\subset Y$, a local modification $\sigma :Z\to W$ with smooth source, and a prime divisor $E$ on $Z$. After shrinking $W$ around the point under consideration, its image is contained in a coordinate neighborhood $U\subset X$. Put $F=\pi \circ \sigma $ and

The map $F$ is holomorphic, and it is injective and biholomorphic off a proper analytic subset; it need not be a proper modification of $U$.

Let $m$ be a positive integer, and let $\nu _ E$ and $\operatorname {ord}_ E$ be the generic Lelong number and the generic vanishing order along $E$, as in section 5.38. The proof of lemma 5.38.9 applies to the map $F$ on relatively compact coordinate patches and gives

In more detail, the upper inequality follows by pulling back the local Bergman lower bound on $U$. For the lower one, a germ $f\in \mathcal I(ma)$ pulls back to a coefficient $(f\circ F)\operatorname {Jac}F$ that is square integrable against $\mathrm e^{-ma\circ F}$, because on a relatively compact patch of $Z$ the change-of-variables integral is bounded by the integral over the image of the patch. Generic divisorial removal and the one-variable integral then give

This estimate only uses the injectivity of $F$ off the exceptional set and the compactness of the patch, not properness of $F$ over $U$. The same estimates hold for $b$.

The hypothesis gives $\mathcal I(ma)\subseteq \mathcal I(mb)$ for every $m>0$, so the orders of the pulled-back ideals satisfy the reverse inequality. Letting $m\to \infty $ in the two estimates proves

This holds for every local divisor over $Y$. Blowing up a point on the source of any local modification recovers the Lelong number at that point as a generic Lelong number along the exceptional divisor; in dimension one the point already is a divisor. The local criterion in theorem 5.37.3 now gives $a\circ \pi \preceq _{\mathcal I}b\circ \pi $. No extension of $\sigma $ beyond its local domain was used. $\square$

Proof. Apply the local order equivalence of lemma 5.43.23 to both $(a,b)$ and $(b,a)$. Together, the two order relations are equivalent to equality of the multiplier ideals at every positive weight. $\square$