5.43 Pullback and descent88AEFA5005B9

Proof. Suppose first that $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$. On a coordinate ball, convolve $u$ and $\theta $ with a smooth nonnegative radial kernel. The resulting smooth functions $u_\epsilon $ satisfy

Write $u=p-r$ locally, with $p$ plurisubharmonic and $r$ smooth. The convolutions $p_\epsilon $ decrease to $p$. On a smaller disk on which $u\circ f$ is not identically $-\infty $, $p_\epsilon \circ f$ therefore converges in $L^1_{\mathrm{loc}}$ to $p\circ f$, while $r_\epsilon \circ f$ converges smoothly. Pulling back the smooth inequality and passing to the limit in the sense of distributions gives $\mathrm{dd}^{\mathrm{c}}(u\circ f)+f^*\theta \geq 0$. Since $u\circ f=p\circ f-r\circ f$ is the sum of a subharmonic and a smooth function, this distributional inequality makes $u\circ f+a$ subharmonic.

Conversely, test on affine complex lines in a coordinate ball. For a fixed nonzero constant direction $\xi $, local integrability of $u$ implies that the restriction of $u$ to almost every line parallel to $\xi $ is not identically $-\infty $. Fubini's theorem, applied to nonnegative smooth test functions, then gives positivity of the scalar distribution obtained by evaluating the Hermitian coefficient matrix of $\theta +\mathrm{dd}^{\mathrm{c}}u$ on $\xi $. Positivity for a countable dense set of constant directions proves positivity of the Hermitian matrix of coefficient measures, which is the current inequality. $\square$

Proof. Fix a Hermitian metric on $M$. The orthogonal complement of $\ker \mathrm dp$ gives a smooth horizontal lift of tangent vectors on $Y$. Around $y_0\in Y$ choose a coordinate ball $B$ centered at $y_0$, with compact closure inside a larger coordinate ball. For $y\in B$, lift the straight segment from $y_0$ to $y$. The ordinary differential equation for horizontal transport has a solution along the whole segment, since its projection remains in the compact closure of $B$, whose inverse image is compact. Smooth dependence on the starting point and on $y$, together with transport along the reversed segment, gives a smooth trivialization

over $B$.

The compact manifold $p^{-1}(y_0)$ has finitely many connected components. The trivialization partitions $p^{-1}(B)$ into finitely many open and closed pieces, each of which meets every fiber in one connected component. On overlaps, two such local partitions identify the same fiber components, and their labels differ by locally constant permutations. Glue one copy of each base ball for every component, using these identifications. Equivalently, take the quotient of $M$ that identifies points in the same connected component of a fiber. The local partitions make this quotient a Hausdorff covering space: distinct points over one base point lie on different sheets, and points over different base points are separated by their images in $Y$.

Call the quotient $Y'$ and its projection $r$. Coordinates of $Y$ on each sheet give holomorphic charts with holomorphic transition maps, so $r$ is a local biholomorphism. It is a finite covering, and $Y'$ is compact because a finite covering of a compact space is compact. Locally, the quotient map $\pi $ is $p$ restricted to one of the open pieces, so it is a holomorphic submersion with the asserted connected fibers. Surjectivity of $\pi $ and connectedness of $M$ make $Y'$ connected. $\square$

Proof. On a coordinate ball $V\subset X$, choose a smooth function $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \theta $. If $u\in \mathrm{PSH}(Y,\pi ^*\theta )$, then $u+\rho \circ \pi $ is plurisubharmonic on $\pi ^{-1}(V)$. Its restriction to a fiber is plurisubharmonic or identically $-\infty $, and since the fiber is compact and connected, the maximum principle shows that it is constant in either case. As $\rho \circ \pi $ is constant on fibers, $u=v\circ \pi $ for a uniquely defined function $v$ on $X$.

A holomorphic submersion admits local holomorphic sections. Composing $u+\rho \circ \pi $ with such a section shows that $v+\rho $ is plurisubharmonic or identically $-\infty $ on a smaller ball. The latter alternative would make $u=-\infty $ on a nonempty open set, which is impossible for a quasi-plurisubharmonic function. Hence $v$ is quasi-plurisubharmonic. Distributional differentiation in local product coordinates gives

Positivity of a pulled-back current of this form implies positivity on the base: test in product coordinates against products with a nonnegative compactly supported volume form of integral one in the fiber variables. It follows that $v\in \mathrm{PSH}(X,\theta )$.

Conversely, by lemma 5.43.2, the pullback of a $\theta $-plurisubharmonic function by $\pi $ is $\pi ^*\theta $-plurisubharmonic; in local product coordinates it is clearly not identically $-\infty $. The pullback map is injective because $\pi $ is surjective. $\square$

Proof. The critical set is a proper compact analytic subset of $X$. It has finitely many irreducible components, and $p$ is constant on each: its differential vanishes on the smooth locus of that component, and continuity extends the constant to the component. Thus the set $D\subset C$ of critical values is finite.

Let $U\in \mathrm{PSH}(X,p^*\beta )$. On a base coordinate disk choose a smooth $a$ with $\mathrm{dd}^{\mathrm{c}}a=\beta $. Over $C\setminus D$, the function $U+a\circ p$ restricts to a plurisubharmonic function, or the constant $-\infty $, on each compact connected fiber. The maximum principle makes it constant there. Local holomorphic sections then give a function $v$ with $U=v\circ p$ and $a+v$ subharmonic. This is the local argument of proposition 5.43.5; it uses compactness of the fibers, not compactness of the punctured total space.

The function $v$ is bounded above by $\sup _ XU$. The removable singularity theorem extends $a+v$ subharmonically across the finitely many punctures. The extensions glue, giving $v\in \mathrm{PSH}(C,\beta )$. Its pullback is $p^*\beta $-plurisubharmonic by lemma 5.43.2. It agrees with $U$ off the proper analytic set $p^{-1}(D)$, hence almost everywhere and everywhere by lemma 5.24.3.

Conversely pullback preserves the curvature inequality by the disk criterion. It is not identically $-\infty $, because $p$ is surjective. Surjectivity also proves uniqueness and preserves the supremum and every inequality $v\leq w+c$. $\square$

Proof. A smooth correction of $v$ pulls back to a smooth function, so we may assume that $v$ is subharmonic. The Taylor expansion of $f$ gives $|f(z)|\leq M|z|^ e$ on a small ball. Choose a unit vector $\xi $ on which the leading homogeneous term is nonzero. Then

For all small $r$, the remainder has modulus at most $|a|r^ e/4$ on $|\zeta |=r$. If $|t|<|a|r^ e/4$, Rouché's theorem shows that $f(\zeta \xi )-t$ has $e$ zeros in $|\zeta |<r$. Consequently, with $c=|a|/4$,

The suprema of $v$ on these three sets are ordered. Dividing by $\log r^2<0$, the two disk terms both tend to $e\nu (v,0)$ by the definition in section 5.34. The middle term therefore has the same limit, proving the formula. $\square$

Proof. On a smooth curve, local modifications are isomorphisms. By theorem 5.37.3, the base inclusions are therefore equivalent to $\nu (v,s)\geq \nu (w,s)$ for every $s\in C$. Let $\pi $ be any local modification of a neighborhood in $X$ with smooth source, and let $y$ be a point of its source. In a coordinate at $s=p(\pi (y))$, the germ $p\circ \pi -s$ is nonconstant. Indeed, a modification is an isomorphism on a dense open set, and a nonconstant holomorphic map cannot be constant on an open set. Its order $e_ y$ is positive and finite. The preceding lemma gives

The base inequalities thus imply all inequalities upstairs, and the same valuative theorem gives the ideal inclusions. Conversely the upstairs inclusions give the inequalities for the identity modification at any $x\in p^{-1}(s)$. Dividing by $\operatorname {ord}_ x(p-s)>0$ recovers the base inequalities. Apply this argument in both directions for equality. $\square$

Proof. Put $M_ j=\sup _ Cv_ j$ and $M=\sup _ Cv$. Hartogs' lemma applied at moving maximum points gives $\limsup M_ j\leq M$. A subsequence with limiting suprema below $M$ would give $v$ below that limit almost everywhere, hence everywhere, a contradiction. Thus $M_ j\to M$.

By surjectivity, $(v_ j-M_ j)\circ p$ has supremum zero. Normalized compactness, theorem 5.25.2, makes these pullbacks precompact in $L^1(X)$. On every relatively compact product chart over $C\setminus D$, where $D$ is the finite set of critical values, Fubini's theorem and bounded smooth volume densities give local $L^1$ convergence to $(v-M)\circ p$. Every subsequential limit therefore equals this pullback off $p^{-1}(D)$, and hence almost everywhere, because that analytic set has smooth volume zero. All subsequential limits agree, so the entire normalized sequence converges. Adding $M_ j\to M$ proves the assertion. $\square$

Proof. Composing a local analytic presentation with $p$ proves the forward implication. The pulled-back generators are not all identically zero, since a nonconstant map to a curve is open.

Conversely fix $s\in C$ and $x\in p^{-1}(s)$. In coordinates, choose a complex line through $x$ on which $p-s$ has its finite order $e$ and is nonconstant, as in the Lelong lemma. On that line write

where $h$ is a local biholomorphism fixing zero; take an analytic $e$th root of the nonvanishing factor of the germ. The restriction of $v\circ p$ to the line is not identically $-\infty $, since $v$ is finite almost everywhere and this map is open. By proposition 5.34.1, it has analytic singularities. In one variable its presentation is $r\log |\zeta |^2+O(1)$ with $r\in \mathbb Q_{\geq 0}$, obtained by taking the least order of its finitely many generators. Changing to $w=h(\zeta )$ gives

Every sufficiently small $t$ has an $e$th root $w$, so $v(s+t)=(r/e)\log |t|^2+O(1)$. This is an analytic presentation on the base, including a bounded one when $r=0$. $\square$

Proof. We first construct the covering over the regular values, complete it across the critical values, and then extend its component quotient to $X$.

The critical values form a finite set $D\subset C$, by the critical-set argument in the proof of 5.43.7; that argument does not use connectedness of the fibers. Over $C^\circ =C\setminus D$, the map $p$ is a proper submersion. The local construction in lemma 5.43.4 applies there: horizontal transport over a disk has compact inverse image over its closure. Compact fibers have finitely many components, and the resulting local labels are glued by locally constant permutations. We obtain

with $q^\circ $ a holomorphic submersion with connected fibers and $r^\circ $ a finite unramified covering of degree $d$. The complement of a proper analytic subset in a connected complex manifold is connected, so $X^\circ $ and its image $C'^\circ $ are connected. Here only the fiber divisor is removed. Locally, the complement of the zero set of a nonzero holomorphic function in a ball is path connected: join two outside points through an outside intermediate point by two complex-line disks. Each restriction is nonzero because it contains an outside endpoint, and its isolated zeros can be avoided. Applying this local argument along a chain of balls proves the required global connectedness.

For each $s\in D$, take a small coordinate disk centered at $s$ and disjoint from the other critical values. A connected component of the covering over its puncture is a finite connected covering of a punctured disk, of some degree $e$. It has a coordinate $w$ in which $r^\circ (w)=w^ e$: the universal cover of the punctured disk is a half-plane with projection $z=\exp \zeta $, and the index-$e$ subgroup of its deck group is generated by $\zeta \mapsto \zeta +2\pi \mathrm i e$. The coordinate $w=\exp (\zeta /e)$ gives the asserted form. Attach one point $w=0$ to each such component. There are finitely many, with their degrees summing to $d$. Gluing these disks to $C'^\circ $ gives a connected Riemann surface $C'$ and a finite holomorphic map $r:C'\to C$. The new points have disjoint disk neighborhoods, so this gluing is Hausdorff. The inverse image of a small closed base disk is a finite union of closed disks; over the complement of their interiors, the original finite covering is proper. Thus $C'$ is compact.

We extend $q^\circ $. Fix $x\in p^{-1}(s)$, $s\in D$. On a small coordinate ball $B$ about $x$, the image of $p$ lies in the chosen disk. The complement $B\setminus p^{-1}(s)$ is connected. Its image under $q^\circ $ therefore lies in a single punctured disk just attached to $C'$, since those disks are the connected components of $r^{-1}$ of the punctured base disk. In its coordinate $w$, the function $w\circ q^\circ $ is holomorphic and bounded, and

The bounded holomorphic removable-singularity theorem extends it across $p^{-1}(s)\cap B$. To see this without assuming that the divisor is smooth, first extend at its smooth points by the one-variable theorem in transverse coordinates. The singular locus of the reduced divisor has complex codimension at least two in the ambient manifold, so Hartogs extension across that locus completes the extension. In dimension one there are only isolated points. Its $e$th power remains $p-s$ by the identity theorem, and hence its value is zero on that fiber. This defines a holomorphic map from $B$ to the attached disk. The local extensions agree on their dense common regular locus and therefore glue to $q:X\to C'$ with $p=r\circ q$. Its image is closed, by compactness, and contains the dense set $C'^\circ $, so $q$ is surjective.

Every fiber of $q$ is connected. Otherwise write one fiber as a union of two disjoint nonempty compact subsets and put them in disjoint open neighborhoods $U_1,U_2$. Properness gives a disk $V$ about their base point $t$ such that

The map $q$ is open: at any point, a complex line on which the nonconstant germ $q-t$ has its first nonzero term gives an open one-variable restriction. Consequently both $q(U_ i)$ contain a neighborhood of $t$. A nearby point of $C'^\circ $ then has a fiber meeting both neighborhoods, contradicting connectedness of the regular fibers.

The fibers of $p$ are the disjoint union of these connected $q$-fibers over the finitely many points of $r^{-1}(s)$. They are its components. Locally at $t$, the formula for a finite map is $r(w)-s=w^{e_ t}$ after changing the coordinate by a holomorphic root of a nonvanishing factor. Pulling back this equality by $q$ gives the divisor multiplicities stated above. Over $C^\circ $, $r$ is unramified and each sheet is one component, proving the degree assertion.

Finally, any factorization with the stated properties has over $C^\circ $ precisely this covering of fiber components. It is unramified there: a critical point of its finite map, together with surjectivity of its first factor, would make $p$ critical over that base point. Thus the component identification is holomorphic on $C'^\circ $. On every punctured disk it identifies two completions of the same finite covering, and the coordinates $w^ e$ above extend it biholomorphically across the added point. It gives a global isomorphism; density and surjectivity make it unique and give commutation with both maps. $\square$

Proof. On a base disk choose a smooth function $a$ with $\mathrm{dd}^{\mathrm{c}}a=\beta $. Off the branch values, $m_ h+a$ is the maximum of finitely many subharmonic functions obtained from $h+a\circ r$ by inverse branches. It is subharmonic there and locally bounded above. The removable-singularity theorem extends it across the branch values. This extension has the stated value: upper semicontinuity and properness give the upper bound at the fiber, while the genuine upper limits of $h+a\circ r$ at its points give the reverse bound. Thus the extensions glue. The function is not identically $-\infty $, since $h$ is finite at some point. This proves the curvature inequality and $h\leq m_ h\circ r$.

Choose coordinates in the disjoint disks over $s$ so that the map on the disk centered at $x$ is $t=z^{e_ x}$. Then

Divide by $\log \delta ^2<0$ and use the supremum definition of Lelong number. Each quotient on the right tends to $\nu (h,x)/e_ x$, and the maximum becomes the minimum after division. Smooth local corrections do not affect these limits.

An ordinary bound $h\leq v\circ r+c$ gives $m_ h\leq v+c$ on every fiber. On a curve, the all-weight ideal order is equivalent to the pointwise Lelong inequalities, by theorem 5.37.3. The pullback formula 5.43.8 makes $h\preceq _{\mathcal I}v\circ r$ equivalent to $\nu (h,x)\geq e_ x\nu (v,r(x))$ at every $x$. The minimum formula gives the required base inequalities. $\square$

Proof. The trace of $g\circ r$ is $dg$, giving the pushforward identity. The same weighted fiber formula for Borel test functions follows from the monotone-class theorem and gives the restriction and atom formulas. The positive and negative Jordan parts of a signed measure have disjoint Borel carriers; their pullbacks are positive with disjoint inverse-image carriers. Each has $d$ times its original mass, which proves the total-variation identity.

Choose a local smooth $a$ with $\mathrm{dd}^{\mathrm{c}}a=\beta $; then $v+a$ is a subharmonic potential of $T_ v$. Its Riesz representation is a logarithmic potential plus a harmonic function. In coordinates $t=z^ e$, the identity

shows that pulling back this potential gives the weighted measure above. Fubini's theorem applies locally: factorization of $z^ e-t$ into its $e$ linear factors gives a uniform local $L^1$ bound for these logarithmic kernels as $t$ varies on a smaller disk. The harmonic term has zero Hessian after pullback. This proves $T_{v\circ r}=r^*T_ v$. The pole set of $v\circ r$ is exactly $r^{-1}\{ v=-\infty \} $, so the restriction formula proves the polar identity.

Inverse images of polar sets are polar by composition of local subharmonic witnesses. For images, take local witnesses on the finitely many inverse-image disks of a small base disk. Off the branch values, the sum of their values over the inverse branches is subharmonic and is $-\infty $ wherever any witness has a pole. It is locally bounded above and is not identically $-\infty $. Its extension across the branch values is subharmonic. Adding the finitely many branch values to the resulting polar set still gives a polar set. This proves the image assertion locally, hence globally.

The pullback of a nonatomic measure has no atoms by the atom formula. The pushforward of a nonatomic measure has no atoms because every fiber is finite. Atomic measures pull back and push forward to atomic measures. Splitting a positive measure into these two parts proves the last assertion. $\square$

Proof. Let $S\subset X$ be a proper analytic subset over whose complement $\pi $ is an isomorphism, and put $D=\pi ^{-1}(S)$. By lemma 5.43.2, pullback preserves the curvature inequality, and the pullback of a $\theta $-plurisubharmonic function is not identically $-\infty $, since $\pi $ is an isomorphism off $D$. The pullback map is injective because $\pi $ is surjective.

To prove surjectivity, let $b\in \mathrm{PSH}(Y,\pi ^*\theta )$, and define $u=b\circ \pi ^{-1}$ off $S$. On a coordinate neighborhood in $X$, choose a smooth function $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \theta $. Then $b+\rho \circ \pi $ is plurisubharmonic on the inverse image of this neighborhood. By properness, its local upper bounds give a local upper bound for $u+\rho $ near $S$, so the removable-singularity theorem extends $u+\rho $ plurisubharmonically across $S$. The resulting quasi-plurisubharmonic extensions of $u$ agree almost everywhere on overlaps, hence everywhere, and glue to a function on $X$.

The current $\theta +\mathrm{dd}^{\mathrm{c}}u$ is positive off $S$. Locally it is the sum of the positive current $\mathrm{dd}^{\mathrm{c}}(u+\rho )$ and the smooth form $\theta -\mathrm{dd}^{\mathrm{c}}\rho $. The smooth form gives $S$ zero mass, so the restriction of $\theta +\mathrm{dd}^{\mathrm{c}}u$ to $S$ is positive as well. Thus $u\in \mathrm{PSH}(X,\theta )$. The plurisubharmonic functions $b+\rho \circ \pi $ and $(u+\rho )\circ \pi $ agree off $D$, hence almost everywhere, hence everywhere. Therefore $b=u\circ \pi $. $\square$

Proof. By lemma 5.25.1, the functions $v_ j$ are uniformly bounded above on relatively compact subsets of $X$. The pullbacks are therefore locally uniformly bounded above and have the common smooth lower curvature bound $-\pi ^*\theta $. After a smooth local correction, the compactness theorem for plurisubharmonic functions shows that every subsequence has a further subsequence that either converges in $L^1_{\mathrm{loc}}$ or tends locally uniformly to $-\infty $ on each connected coordinate region in $Y$.

Every such region meets the complement of the exceptional locus. There $\pi $ is a biholomorphism, and change of variables gives $L^1_{\mathrm{loc}}$ convergence to $v\circ \pi $, which excludes convergence to $-\infty $. Each subsequential limit agrees with $v\circ \pi $ off the exceptional locus, hence almost everywhere, since this analytic set has smooth volume zero. Thus the limit is unique, and applying this argument on a countable cover by coordinate regions proves convergence of the whole sequence in $L^1_{\mathrm{loc}}(Y)$. $\square$

Proof. Choose a proper analytic subset $S\subset X$ such that $\pi $ is an isomorphism outside $D=\pi ^{-1}(S)$. Both $S$ and $D$ have capacity zero. For $\epsilon >0$, choose an open neighborhood $V$ of $D$ of capacity less than $\epsilon $. By properness, there is a neighborhood $U_0$ of $S$ with $\pi ^{-1}(U_0)\subset V$; shrink it to an open set $U$ with $\operatorname {Cap}_ X(U)<\epsilon $. The inverse of $\pi $ is biholomorphic near the compact set $X\setminus U$. A finite cover by nested coordinate charts and the comparison of local and global capacities in section 5.29 give a constant $C_\epsilon $ such that, for every Borel set $E\subset X$,

Apply these inequalities to $E=\{ |v_ j-v|>\delta \} $ for each fixed $\delta >0$, omitting the pluripolar pole sets, and let first $j\to \infty $ and then $\epsilon \downarrow 0$. $\square$

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. $\square$

Proof. For each $m$, choose $y\in Y$ with $\nu (u,y)<1/m$. Skoda's local integrability criterion gives $\mathcal I(mu)_ y=\mathcal O_{X,y}$, so the image ideal $\mathcal I(mu)\mathcal O_ Y$ does not vanish identically. Since $u_ m$ has analytic type $(1/m,\mathcal I(mu))$, proposition 5.34.1 shows that $u_ m|_ Y$ is quasi-plurisubharmonic on $Y$. Its pullback to $Z$ is quasi-plurisubharmonic as well, since $\pi $ is biholomorphic off a proper analytic subset.

Restricting and pulling back the curvature inequality of corollary 5.36.2, which is allowed by lemma 5.43.2, gives

Since the errors $\epsilon _ m$ are bounded, a sufficiently large multiple of a single Hermitian form on $Z$ dominates $\pi ^*(\theta |_ Y)+\epsilon _ m\pi ^*(\omega |_ Y)$ for all $m$. Lemma 5.38.8, applied on $Z$ in this fixed background, therefore bounds the generic Lelong numbers along $E$ uniformly in $m$. $\square$