Lemma 5.43.22599D8A3D42BC

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. $\square$