Lemma 5.43.236AA2493173A9

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. $\square$