Lemma 20.3.44C53A2B3087AB

Proof. We first extract a divisor on the base and then check the effect of ramification on its generic Lelong number. Put $r=\dim M$. Maintain a compact analytic graph proper over $Z$, isomorphic to $Z$ near a general point of $D$, with a dominant map to the current smooth modified base. Its divisor image $S$ is irreducible analytic by the proper mapping theorem. The graph may be singular away from this neighborhood.

Near a general point of $D$, let $J_ D(f)$ be the least vanishing order of the coefficients of $f^*(\mathrm dy_1\wedge \cdots \wedge \mathrm dy_ r)$ in a smooth source frame. Dominance makes it a finite nonnegative integer. Changes of smooth coordinates multiply the target volume form by a unit and change the source frame by an invertible matrix, so this integer is independent of the coordinates.

Suppose $S$ has codimension $c\geq 2$. Blow up its reduced coherent ideal. Over the smooth part of $S$, use coordinates with $S=\{ y_1=\cdots =y_ c=0\} $ and put

Choose $y_1$ attaining this minimum. After removing a proper analytic subset of $D$, its leading coefficient is a unit, and all ratios $f^*y_ i/f^*y_1$ are holomorphic. The map therefore lifts near these points to the smooth blowup chart

Resolve the target blowup by a proper analytic resolution which is an isomorphism over its whole smooth locus; the resolution input is recalled in 25.15. The resulting base is smooth and compact, and the lift near the general point of $D$ is preserved. Taking the main analytic graph of the resulting meromorphic lift again gives a compact graph proper over $Z$, unchanged in this neighborhood.

In these coordinates,

Consequently the new Jacobian order is $J_ D(f)-(c-1)m\geq 0$. It strictly decreases at every step. After finitely many steps the divisor image has codimension one: its image cannot dominate a modified base when its original image was proper. This proves extraction without resolving the source graph in a way that changes the general point of $D$.

Let $e$ be the order along $D$ of the pullback of a local equation of the extracted divisor $F$. Near its general point, the ideal $\mathfrak b\mathcal O_{M'}$ is a power of that equation times a unit ideal. Pullback gives the ideal-order identity in (20.3.45).

The map $D\to F$ has tangential rank $r-1$ at general smooth points. Use the tangential base coordinates as source coordinates and absorb the unit in the normal pullback into a local holomorphic $e$th root. The resulting local form of the map is

After a smooth local correction, Siu's divisorial removal from section 5.38 writes

with $R$ plurisubharmonic and $\nu (R,F)=0$. By the non-strict Lelong level theorem in 5.34.2, its full point Lelong number is zero at a very general point of $F$. Skoda's theorem then makes $\mathrm e^{-tR}$ locally integrable for every $t>0$ at this point.

For $e>1$, choose $1<p<e/(e-1)$ and let $p'$ be its conjugate exponent. The density $|y_1|^{2/e-2}$ belongs locally to $L^ p$. Change of variables and Hölder's inequality, using integrability of $\mathrm e^{-tp'R}$, give

on sufficiently small corresponding polydisks. Free source variables contribute a finite product factor; for $e=1$ Fubini's theorem suffices. A positive generic divisor coefficient $c'$ of the pulled-back residual would instead force nonintegrability for $tc'\geq 1$. The residual coefficient is therefore zero, and the logarithmic term has coefficient $ec_0$. This proves the Lelong identity even if $R|_ F\equiv -\infty $.

Finally, if $f(D)=M$, the map is a submersion near a very general point of $D$. A nonzero ideal is a unit at a general image point, and Skoda's theorem at an image point of zero Lelong number, followed by Fubini's theorem in product coordinates, gives zero generic Lelong number on $D$. $\square$