20.3 Restriction and intrinsic trace dataE6AE4A881F9F

Proof. Fix the modification $\pi :Z\to Y$ carrying $E$, and write $b(t)=\operatorname {ord}_ E\mathfrak a_ t(u;Y)$ for $t>0$. For $s,t>0$, the subadditivity of multiplier ideals (the second inclusion in (5.37.6)), followed by restriction to $Y$, gives

Moreover, $b(t)\geq 0$. For fixed $s>0$ and $t=ks+r$ with $k$ a nonnegative integer and $0\leq r<s$, repeated superadditivity gives $b(t)\geq k b(s)$. Hence

Taking the supremum over $s$ proves the existence of the limit and the supremum formula, with the value $+\infty $ allowed at this stage. In particular, the limit along positive integers is the same.

For every sufficiently large integer $m$, the restriction $u_ m|_ Y$ has analytic type $(1/m,\mathfrak a_ m(u;Y))$ by section 5.34, and therefore

These numbers are uniformly bounded by corollary 5.43.26, so the limit along integers, and hence the supremum, is finite. Finally, the ideals $\mathfrak a_ t(u;Y)$ are unchanged when $u$ is replaced by a potential with the same multiplier ideals, which proves the last assertion. $\square$

Proof. The inclusions of ideals defining $\preceq _{\mathcal I}$ persist after restriction and reverse the vanishing orders, which proves (1). Replacing $t$ by $at$ in (20.3.3) proves (2). Moreover,

Since the order of a sum of ideals is the minimum of their orders, restriction and passage to the limit prove (4); note that $\max (u,v)$ also has zero generic Lelong number along $Y$. Generic Lelong numbers of sums add, so $u+v$ has zero generic Lelong number along $Y$ as well.

For (3), subadditivity of multiplier ideals gives $\tau _{Y,u+v}\geq \tau _{Y,u}+\tau _{Y,v}$. For the reverse inequality, fix a sufficiently large integer $k$ and the analytic approximant $S_ k$ of $u+v$ given by theorem 5.36.1, of analytic type $(1/k,\mathcal I(k(u+v)))$; that theorem gives $u+v\leq S_ k$. Let $U_ m,V_ m$ be the corresponding approximants of $u$ and $v$. Lemma 5.38.12 gives

Here both $\delta _ m$ and the bounded error may depend on the fixed $k$. The restrictions of $U_ m$, $V_ m$ and $S_ k$ to $Y$ are not identically $-\infty $ by corollary 5.43.26. Restricting, pulling back to a fixed modification and taking the generic Lelong number along a divisor $E$ on it, we obtain

Letting first $m\to \infty $, with $k$ fixed, and then $k\to \infty $, the finite limits of theorem 20.3.2 give the reverse inequality. $\square$

Proof. The restriction theorem, which is the first inclusion in (5.37.6), and lemma 5.38.9, applied on $Y$, give for every $t>0$

Divide by $t$ and use theorem 20.3.2.

If $u$ has analytic singularities, apply lemma 5.38.12 to the pair $(u,0)$, with $s=u$, $U_ m=u_ m$ and $V_ m=0$. It gives

Restricting to $Y$, pulling back by $\pi $ and taking the generic Lelong number along $E$, we obtain, since $u_ m$ has analytic type $(1/m,\mathcal I(mu))$,

Letting $m\to \infty $ proves the reverse inequality, hence equality. All the restrictions involved are quasi-plurisubharmonic, because $\nu (u,Y)=0$. $\square$

Proof. The first equality follows from lemma 18.27.2, lemma 5.37.5. In normal crossing coordinates, the weight for $t\sigma _\Delta $ is a smooth positive factor times $\prod _ j|z_ j|^{-2ta_ j}$. One-variable polar integration gives the exact formula

Indeed, integrability of a germ of transverse order $k$ requires $k+1>ta_ j$, so $k\geq \lfloor ta_ j\rfloor $; equality at the preceding order is divergent when $ta_ j$ is an integer. Necessity follows on an open set of the other variables where the leading coefficient is nonzero. After division by these powers, the remaining product weight is integrable, proving sufficiency.

Since $Y$ is contained in no $D_ j$, each restricted defining equation is a nonzero holomorphic germ on $Y$. Its pullback to $Z$ is nonzero and has a finite nonnegative order $N_{j,E}$ along $E$. Restriction of the displayed principal ideal and then pullback therefore give the exact order $\sum _ j\lfloor ta_ j\rfloor N_{j,E}$. Dividing by $t$ proves the trace formula, including along all real weights. Moreover $\nu (\sigma _\Delta +q,Y)=0$: outside the finitely many divisors both summands have zero point Lelong number. Thus these orders are the intrinsic data of theorem 20.3.2. When $\Delta =0$, the ideal is $\mathcal O_ X$ and every trace is zero. $\square$

Proof. For integer $m$ let

A nonzero coherent ideal on a compact smooth curve has finitely many zeros, so this sum is finite. Since $u_ m|_ Y$ has analytic type $(1/m,\mathfrak a_ m(u;Y))$, the measure $D_ m$ is the atomic part of the positive current $(\theta +\epsilon _ m\omega )|_ Y+\mathrm{dd}^{\mathrm{c}}(u_ m|_ Y)$, and Stokes' theorem on the curve gives

For any finite subset of $Y$, passing to the limit in the sum of the corresponding coefficients by theorem 20.3.2, and then taking the supremum over finite subsets, gives (20.3.11). In particular, the positive coefficients have at most countable support and define a finite measure. Write $d=D_ u(Y)$.

Choose a Green kernel $G$ for $\mathrm{dd}^{\mathrm{c}}$ on the curve, normalized so that

Such a kernel exists: the Green kernel of the scalar Laplacian on the compact Riemann surface has, in a local coordinate $z$, the singularity $\log |z(x)-z(p)|^2$, it is uniformly bounded above, and $\sup _ p\int _ Y|G(x,p)|\eta (x)<\infty $; subtracting a constant gives the sign condition. By Fubini's theorem,

is integrable, and it is quasi-subharmonic with $\mathrm{dd}^{\mathrm{c}}w=D_ u-d\, \eta $. Its Lelong number at every point $p$ is the mass of $D_ u$ at $p$, namely $\tau _{Y,u}(p)$.

Since $M\eta -\theta |_ Y$ is smooth with integral zero, the Poisson equation gives a smooth $s$ with $\mathrm{dd}^{\mathrm{c}}s=M\eta -\theta |_ Y$. Then $v=w+s$ satisfies (20.3.12), and the right-hand side is positive because $M\geq d$. At every atom of $D_ u$, the positive Lelong number forces $v=-\infty $. The extended measure of $v$ is the restriction of the current in (20.3.12) to the set where $v>-\infty $; this set contains no atom of $D_ u$, and its complement is polar and therefore not charged by $\eta $. Hence the extended measure is $(M-d)\eta $.

If another smooth form $\eta $ or another smooth background $\theta $ is used, the two constructed representatives have the same atomic curvature, and all other terms in their equations are smooth. Their difference therefore has smooth $\mathrm{dd}^{\mathrm{c}}$, so it is smooth by elliptic regularity, and in particular bounded. This proves independence of these choices at the level of ordinary types. Independence of the choice of $u$ within its multiplier-ideal class holds because the numbers defining $D_ u$ depend only on that class. $\square$

Proof. The supremum formula in (20.3.3) gives $D_ m\leq D_ u$ coefficient by coefficient. Since the coefficients converge and are dominated by those of the finite measure $D_ u$, dominated convergence gives $D_ m(Y)\to D_ u(Y)$, and hence convergence in total variation. Superadditivity with $s=t=m$ gives $D_{2m}\geq D_ m$, and since $G\leq 0$ the potentials $w_{2^ j}$ decrease. Monotone convergence for $-G$ identifies their pointwise limit with $w$, including at poles. The uniform $L^1$ bound for the kernel also gives

Each $D_ m$ is finitely supported, and locally both $w_ m$ and $u_ m|_ Y$ are the sum of the logarithmic singularities prescribed by $D_ m$ and a bounded function. Thus their ordinary types agree.

If $a=u|_ Y$ is not identically $-\infty $, proposition 20.3.6 gives $\nu (a,p)\geq \tau _{Y,u}(p)$ at every point. Since $\mathrm{dd}^{\mathrm{c}}w=D_ u-D_ u(Y)\eta $, lemma 5.38.5 implies $a\preceq w$. If $u$ has analytic singularities, proposition 20.3.6 identifies the finitely many logarithmic coefficients of $u|_ Y$ with those of $D_ u$, and the bounded local remainders give ordinary equivalence. $\square$

Proof. Assume that $M>d$, take the value of $c$ given above, and solve

The right-hand side has integral zero, so the elliptic $L^ p$ theory gives a solution $s\in W^{2,p}(Y)$, unique up to constants. In real dimension two, the Sobolev embedding for $p>1$ makes $s$ continuous and bounded. The potential $v=w+s$ has the atomic trace type and satisfies

Its extended measure is $c f\, \mathrm{d}V$, since this absolutely continuous measure does not charge the polar set of poles, while the atoms lie in it. For $M=d$, the same construction with $c=0$ has a smooth right-hand side and gives the asserted potential with zero extended measure.

To prove necessity and uniqueness, let $a$ be any $\theta |_ Y$-plurisubharmonic potential in the ordinary type of $w$, and put $\mu =\theta |_ Y+\mathrm{dd}^{\mathrm{c}}a$. By corollary 5.27.15, ordinary equivalence with $w$ makes the polar part of $\mu $ equal to $D_ u$. Consequently

Integrating the equation therefore forces the stated value of $c$, so a positive $c$ requires $M>d$. If two potentials solve the equation with the same density and constant, their curvature currents both equal $D_ u+c f\, \mathrm{d}V$; their difference has zero $\mathrm{dd}^{\mathrm{c}}$ and is therefore constant on the compact connected curve. The same argument proves uniqueness in the case of zero extended measure. $\square$

Proof. The curvature bounds of corollary 5.36.2, restricted to $Y$, are dominated by a single form $A\eta $ once $A$ is large. Since $D_ m\leq D_ u$ and $G\leq 0$, we have $w\leq w_ m$. Also

Hence their extended masses are $A-D_ u(Y)$ and $A-D_ m(Y)$, both positive. For $a,b\in \mathrm{PSH}(Y,A\eta )$ with $a\leq b$, the formula for the distance of ordered potentials [Lemma 6.2.2, SGPT] reads

Substituting $a=w$ and $b=w_ m$ proves the second equality in (20.3.18). Since $u_ m|_ Y$ and $w_ m$ are ordinarily equivalent, their distance vanishes, which proves the first equality. The convergence of $D_ m$ in total variation proves the limit.

Each $w_ m$ has finitely many logarithmic poles with rational coefficients, hence analytic singularities. The characterization in Theorem 18.1.2, applied on the compact Kähler curve $Y$, shows that their $d_ S$-limit $w$ is $\mathcal I$-good.

Finally, suppose that $X$ is Kähler. The Kähler trace is defined in Definition 20.2.6, Proposition 20.2.4 through a quasi-equisingular approximation of $u$ with analytic types $(2^{-j},\mathcal I(2^ ju))$, whose restriction to $Y$ has the ordinary type of $w_{2^ j}$. The same distance calculation shows that these restrictions converge to $w$ for $d_{S,A\eta }$, so $w$ represents the Kähler trace up to $P$-equivalence. $\square$

Proof. One has

Every atom of $D$ is a pole of $w_ D$, and the smooth measure $\eta $ charges no polar set, including any additional poles caused by accumulation of the atoms. Hence $\mathrm{MA}_\beta (w_ D)=(A-D(Y))\eta $.

Put $q=q_ D$. By theorem 18.4.2, $q$ has the same multiplier ideals, and hence the same Lelong numbers at points, as $w_ D$. Applied to $\mathrm{dd}^{\mathrm{c}}w_ D=D-D(Y)\eta $, lemma 5.38.5 gives $q\preceq w_ D$, and the envelope inequality $q\geq w_ D-\sup _ Yw_ D$ proves $q\sim w_ D$. By corollary 5.27.15, the polar part of $\beta +\mathrm{dd}^{\mathrm{c}}q$ is therefore also $D$, and Stokes' theorem gives the non-pluripolar mass $A-D(Y)$.

For any $a\in \mathrm{PSH}(Y,\beta )$, its Lelong numbers at points give an atomic measure $D_ a$ of mass at most $A$. On a smooth curve these numbers determine all multiplier ideals, by theorem 5.37.3. Thus $a\sim _{\mathcal I}w_{D_ a}$, so $P_\beta [a]_{\mathcal I}=q_{D_ a}$. If $D_ a(Y)=A$, the atoms exhaust the whole curvature measure, and the non-pluripolar mass is zero. This proves surjectivity and the description of the potentials of positive mass. Distinct atomic measures give distinct Lelong numbers, which proves injectivity. $\square$

Proof. Write $d_ S=d_{S,\beta }$ and $a\lor b=\max (a,b)$. For potentials $u,v\in \mathrm{PSH}(Y,\beta )$ of positive mass, besides the formula for ordered potentials we use the inequality [Lemma 6.2.1, SGPT]

on a curve; here $48=3(1+1)2^{1+2}$ is the constant of that lemma in dimension one.

Put $z=\max (w_ D,w_ E)$. By lemma 20.3.20, the polar parts of the curvature measures of $w_ D$ and $w_ E$ are $D$ and $E$. By (5.27.16), the polar part of the curvature measure of $z$ is $D\wedge E$, the atomic measure with the smaller coefficient at each point. Stokes' theorem therefore gives

The formula for ordered potentials yields

Combining this equality with the two inequalities above proves both bounds, since replacing $w_ D,w_ E$ by the ordinarily equivalent $q_ D,q_ E$ does not change the singularity distance. The lower bound and the injectivity in lemma 20.3.20 make the pseudometric a metric on these envelopes. $\square$

Proof. By lemma 20.3.20, the space corresponds to the positive atomic measures of mass at most $A-\varepsilon $. This set is complete for total variation. Indeed, a Cauchy sequence has as limit a positive finite measure with the same mass bound. The union of the countable supports of the sequence is countable, and total-variation convergence shows that the limit gives zero mass to the complement of that union, so the limit is again atomic. The two metric estimates of theorem 20.3.21 transfer completeness to the ideal envelopes.

Finite truncation, followed by rational approximation of each coefficient from below, preserves the mass bound and converges in total variation. The resulting Green potentials have finitely many logarithmic poles with rational coefficients, and their ideal envelopes have the same ordinary types by lemma 20.3.20. These envelopes therefore have analytic singularities, and the upper metric estimate proves density.

Finally, fix $0<a<A-\varepsilon $. The measures $a\delta _ p$, for distinct $p\in Y$, have pairwise total-variation distance $2a$, so their ideal envelopes have pairwise $d_ S$ distance at least $a/48$. This infinite separated family rules out total boundedness. $\square$

Proof. The restrictions $u_ m|_ Y$ have a common smooth lower bound for their curvature. After subtracting their suprema, normalized compactness gives a subsequence converging in $L^1$ to some $v$. The inequalities $(\theta +\epsilon _ m\omega )|_ Y+\mathrm{dd}^{\mathrm{c}}(u_ m|_ Y)\geq 0$ pass to the limit as $\epsilon _ m\to 0$, giving $\theta |_ Y+\mathrm{dd}^{\mathrm{c}}v\geq 0$.

For each fixed compact modification $\pi :Z\to Y$, lemma 5.43.19 gives $L^1$ convergence of the pullbacks along the same subsequence, and lemma 5.38.7 gives, at every point $z$ of a prime divisor $E\subset Z$,

Taking the generic value along $E$ proves the assertion. No diagonal extraction over divisors is needed, since the pullback lemma applies to each modification once the subsequence on $Y$ has been fixed. $\square$

Proof. Step 1: $T$ is the largest trace competitor. The argument of proposition 20.3.24, applied with the assumed curvature bounds, gives a normalized $\Omega $-plurisubharmonic limit $v$ of a subsequence of the $a_ m$, whose divisorial Lelong numbers dominate $\tau _{Y,u}$. Hence the family defining $T$ is nonempty, and its supremum lies between $v$ and zero. The family is stable under finite maxima, since the Lelong number of a maximum is the minimum of the two Lelong numbers, also after pullback. By Choquet's lemma and finite maxima, there is an increasing sequence of trace competitors whose limit has upper semicontinuous regularization $T$. Convergence of the pullbacks and semicontinuity of Lelong numbers show that $T$ satisfies every defining inequality. Thus $T$ is itself a trace competitor, and it is the largest one.

Step 2: a common lower barrier. Fix $m$, and let $\pi :Z\to Y$ be a log resolution of $\mathfrak a_ m(u;Y)$. For each prime component $E$ of the resulting divisor, (20.3.3) gives

The criterion of lemma 5.38.11, which involves only these finitely many divisors, gives $T\preceq a_ m$. Since $T\leq 0$ and $\delta _ m\geq 0$, the function $T$ is a competitor for $q_ m$. Moreover, proposition 18.4.8 identifies the two envelopes in (20.3.28) and gives $q_ m\sim a_ m$. Therefore

for every divisor over $Y$.

Step 3: convergence. All $q_ m$ have supremum zero and a common smooth lower bound for their curvature. Let $q$ be any $L^1$ cluster point. It belongs to $\mathrm{PSH}(Y,\Omega )$, lies between $T$ and zero, and satisfies

along the convergent subsequence, by the pullback and semicontinuity argument of proposition 20.3.24 on each fixed compact modification. Thus $q$ is a trace competitor, so $q\leq T$, and hence $q=T$. Normalized compactness now gives $L^1$ convergence of the whole sequence.

Step 4: monotonicity and the fixed-point identities. Subadditivity gives $\mathfrak a_{2m}(u;Y)\subseteq \mathfrak a_ m(u;Y)^2$, hence $a_{2m}\preceq a_ m$. When the backgrounds $\Omega +\delta _{2^ j}h$ decrease, this implies $q_{2^{j+1}}\leq q_{2^ j}$. The lower bound by $T$ shows that the pointwise limit is not identically $-\infty $, and since it agrees with $T$ almost everywhere by $L^1$ convergence, it equals $T$ everywhere. The family defining $T$ depends only on $\Omega $ and on the multiplier ideals of $u$, which proves the asserted independence.

Finally, $P_\Omega [T]_{\mathcal I}$ has the same multiplier ideals as $T$, hence the same divisorial Lelong numbers, so it is a trace competitor. Maximality and the inequality $P_\Omega [T]_{\mathcal I}\geq T$ give $P_\Omega [T]_{\mathcal I}=T$, and the identity $P_\Omega [T]=T$ follows from corollary 18.4.4. $\square$

Proof. For any nonempty family of trace competitors, the argument with finite maxima and semicontinuity in Step 1 of the proof of theorem 20.3.27 shows that the envelope is its largest member. By proposition 5.43.18, every $\pi ^*\Omega $-plurisubharmonic function is uniquely of the form $a\circ \pi $ with $a\in \mathrm{PSH}(Y,\Omega )$, and since $\pi $ is surjective, $a\leq 0$ if and only if $a\circ \pi \leq 0$. If $a$ satisfies all trace inequalities over $Y$, then $a\circ \pi $ satisfies those over $Z$, since every modification of $Z$, composed with $\pi $, is a modification of $Y$.

Conversely, suppose that $a\circ \pi $ satisfies the inequalities over $Z$. Let $E$ be a prime divisor on a smooth compact modification $V\to Y$. Resolving the graph of the bimeromorphic map between $V$ and $Z$ gives a smooth compact modification $W$ dominating both. The map $W\to V$ is an isomorphism near the generic point of $E$, so the strict transform $F$ of $E$ has the same ideal orders and generic Lelong numbers as $E$, and the inequality at $F$ over $Z$ is exactly the required inequality at $E$. Thus the two families of competitors correspond under pullback, and the largest member on $Z$ is $T\circ \pi $. This proves the first identity.

For $\varepsilon >0$, the pullback $T\circ \pi $ is still a competitor for $\mathcal T^\pi _{\pi ^*\Omega +\varepsilon h}(u;Y)$. These envelopes lie between $T\circ \pi $ and zero and decrease as $\varepsilon \downarrow 0$, so their limit $S$ satisfies $S\geq T\circ \pi $ and is $\pi ^*\Omega $-plurisubharmonic. The $L^1$ convergence of pullbacks and semicontinuity of Lelong numbers preserve every trace inequality, so $S$ belongs to the unperturbed family. Maximality gives $S\leq T\circ \pi $, hence equality. Since the lower bound $T\circ \pi $ is integrable on $Z$, dominated convergence gives $L^1$ convergence.

For the final assertion, denote the analytic envelope in the statement by $q_ m$. The proof of theorem 20.3.27 applies to the global coherent ideals $\mathfrak a_ m(u;Y)\mathcal O_ Z$: resolving each of them and applying lemma 5.38.11 gives $T\circ \pi \preceq b_ m$. Hence

for every divisor $F$ over $Z$, where ordinary and ideal envelopes agree by proposition 18.4.8. Every $L^1$ cluster point is $\pi ^*\Omega $-plurisubharmonic and satisfies all trace inequalities, so the first identity in (20.3.31) identifies it with $T\circ \pi $, and compactness gives convergence of the whole sequence. Subadditivity orders the dyadic analytic types, decreasing dyadic backgrounds then order the envelopes, and their pointwise limit is the same $L^1$ limit. $\square$

Proof. The curvature errors $\epsilon _ m$ tend to zero, so $u_ m|_ Y\in \mathrm{PSH}(Y,\kappa )$ for all sufficiently large $m$. By theorem 20.3.27, applied with $\Omega =\kappa $ and $\delta _ m=0$, the envelope fixed points

decrease to $T=\mathcal T_\kappa (u;Y)$, after discarding finitely many indices. The normalized potential $v$ of proposition 20.3.24 belongs to $\mathrm{PSH}(Y,\Omega _0)\subset \mathrm{PSH}(Y,\kappa /2)$ and satisfies $v\leq T$, so $P_{\kappa /2}(T)\not\equiv -\infty $. Thus lemma 16.28.69 and Stokes' theorem give, with $r=\dim Y$,

For a decreasing sequence of envelope fixed points in a closed background, positive mass of the limit implies convergence of every divisorial Lelong number [Proposition 3.1.12, SGPT]. Applied on $Y$, this yields

Here the middle equality uses the ordinary equivalence of each $q_ j$ with $u_{2^ j}|_ Y$.

If $\Omega \geq \kappa $, then $T$ is a trace competitor for $\Omega $, so $T\leq T_\Omega =\mathcal T_\Omega (u;Y)$. Together with the defining inequalities of $T_\Omega $, this gives

This proves the assertion for every such $\Omega $. $\square$

Proof. Since $\theta |_ Y+\epsilon _ m\omega |_ Y\leq \Omega $ for all sufficiently large $m$, the restricted approximants belong to $\mathrm{PSH}(Y,\Omega )$. Theorem 20.3.27, applied with $\delta _ m=0$, gives $T$, its defining inequalities and both fixed-point identities.

Normalize $v$ so that $\sup _ Yv=0$. Since $v\in \mathrm{PSH}(Y,\Omega /2)$ is a trace competitor, it lies below $T$ and is a competitor for $P_{\Omega /2}(T)$. The contact estimate for envelope fixed points, proposition 16.28.7, applied on $Y$, gives the bound for the measure and its finiteness, and lemma 16.28.69 gives strict positivity. $\square$

Proof. By theorem 20.3.35, $T$ satisfies both parts of the model condition (16.28.67) for the Hermitian background $\Omega $ on $Y$, so theorem 15.12.1 applies. $\square$

Proof. Lemma 5.37.7 and restriction to $Y$ give

The nonzero coherent ideal $\mathfrak a_ t(U;C)$ on the smooth curve is generated near $b$ by $t_ b^{k_ t(b)}$, where $k_ t(b)=\operatorname {ord}_ b\mathfrak a_ t(U;C)$. For a divisor $E$ with $h(E)=\{ b\} $, the order along $E$ of the extended ideal is therefore $k_ t(b)\operatorname {ord}_ E(t_ b\circ h)$. For a divisor dominating $C$, the generic order is zero, since the ideal on the curve has only finitely many zeros. Dividing by $t$ and letting $t\to \infty $ gives exactly the stated formula for $\tau _{Y,u}(E)$.

For the Lelong numbers of $v\circ \pi $, suppose first that $h(E)=\{ b\} $. At a general point of $E$, a local equation $z_1=0$ of $E$ satisfies $t_ b\circ h=z_1^ k g$ with $g$ holomorphic and nonvanishing, and $k=\operatorname {ord}_ E(t_ b\circ h)$. Replacing $z_1$ by $z_1g^{1/k}$ for a local holomorphic $k$th root, we may assume $t_ b\circ h=z_1^ k$. The Lelong number of the one-variable function $w(z_1^ k)$ at $0$ is $k\nu (w,b)=k\tau _{C,U}(b)$, as follows from the definition through circular means, and this is also the generic Lelong number of $v\circ \pi $ along $E$ in these product coordinates. If $h(E)=C$, the regular locus of $E$ contains a point mapped to a point where $w$ is finite, because the nonconstant holomorphic map from that locus to $C$ has open image; the pullback is finite there, so its generic Lelong number along $E$ is zero. This proves exact realization on every compact model.

By lemma 5.38.5 and (20.3.38), a quasi-plurisubharmonic function $a$ on $Y$ satisfies $a\preceq v$ if and only if $\nu (a,Y_ b)\geq \tau _{C,U}(b)$ for every $b$ with $\tau _{C,U}(b)>0$. These fiber inequalities are among the trace inequalities, and conversely domination by $v$ implies all trace inequalities on modifications, by the exact formula just proved. Thus the trace competitors for $\Omega $ are exactly the competitors for the ordinary envelope of $v$, and $\mathcal T_\Omega (u;Y)=P_\Omega [v]$.

For $\psi =P_\Omega [v]_{\mathcal I}$, the Lelong numbers along the fibers are those of $v$, since $\psi $ and $v$ have the same multiplier ideals, and the same domination lemma gives $\psi \preceq v$. Hence

Since $\psi \leq 0$, it is a competitor for the ordinary envelope of $v$, so $\psi \leq P_\Omega [v]$. This proves (20.3.40).

The potential $v-\sup _ Yv$ is a competitor for $P_{\Omega /2}(P_\Omega [v])$. The contact estimate for envelope fixed points, proposition 16.28.7, gives finite mass, and lemma 16.28.69 gives strict positivity. The two envelope properties give the model condition (16.28.67), so theorem 15.12.1 gives the stated equations.

Two Green constructions on $C$ differ by a smooth function, by the Poisson argument in the proof of theorem 20.3.10, so their pullbacks have the same ordinary type. Every background in the statement gives an envelope ordinarily equivalent to this pullback. This proves the asserted independence, at the level of ordinary types rather than of normalized potentials. $\square$

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 $D$. $\square$

Proof. Apply lemma 20.3.44 to $f=q\circ \pi $. If $D$ dominates $B$, both the trace order and the Lelong number of $q^*w_ a$ are zero. Otherwise the ideal identity (20.3.42) and the extracted divisor $F$ give

Thus the same potential $q^*w_ a$, independent of the choice of $D$, is an exact trace competitor on every compact model.

The compact elliptic group $\mathbb C^*/\langle \mathrm e^{-1}\rangle $ acts by scalar multiplication on $Y$. It preserves $\alpha $, $\beta $ and all ideals in (20.3.42). It also preserves the full competitor family: for a translation $\rho $, the constraint for $v\circ \rho $ on $\pi $ is the constraint for $v$ on the modification $\rho \circ \pi $, with the same ideal order. By theorem 20.3.27, the envelope is the largest member of this family, so it is invariant under the group. The action is transitive on each fiber; local holomorphic sections of $q$ therefore show that the envelope descends to a genuine quasi-plurisubharmonic function $v$ on $B$.

For a basic potential, positivity is equivalent to positivity on the base:

Indeed, wedge the left-hand current with the smooth semipositive form $\beta $ and push forward by $q$. Since $\beta ^2=0$ and its fiber integral is two, the resulting positive current is $2(a\kappa +\mathrm{dd}^{\mathrm{c}}v)$. Conversely, pull back the base inequality and add $b\beta \geq 0$.

Every smooth compact modification $\sigma :B'\to B$ lifts to the smooth compact modification $Y\times _ B B'\to Y$. The inverse image of a prime divisor $F\subset B'$ is prime, because the fibers are connected, and has generic multiplicity one. The trace constraint on this divisor says exactly that $v$ satisfies the constraint at $F$ on $B'$. Thus $v\leq w_ a$. The opposite inequality follows from the exact competitor $q^*w_ a$, proving (20.3.49).

For $\Omega \geq \Omega _{a,b}$, this competitor lies below $T_\Omega =\mathcal T_\Omega (u;Y)$. The defining lower bound and the order of the two potentials give

which proves exactness in every such background. $\square$

Proof. Normalize $v\leq 0$ and put $v_ j=\max (v,-j)$. For bounded potentials, the horizontal current has rank at most $d=n-1$ and $\beta ^2=0$, so the bounded local product gives

This identity follows by local decreasing regularization on base charts and the bounded product convergence of section 13.14; it does not require $\beta $ to be closed. Restrict to $q^{-1}\{ v>-j\} $ and pass to the increasing non-pluripolar measures. Fiber integration gives (20.3.53), and (20.3.47) gives the two bounds. $\square$

Proof. The singular locus of $U$ is the projective line defined by $z_0=\cdots =z_{n-2}=0$, of codimension $d$. Its smooth-center multiplier ideals are given by lemma 5.37.1; restriction to $B$ gives the displayed powers of $\mathfrak m_ P$. Dividing their orders by $t$ proves the trace formula. Exactness of the ordinary restriction follows from proposition 20.3.6, since the ambient pullback has analytic singularities.

On the point blowup $\sigma :\widetilde B\to B$, let $F$ be the exceptional divisor, $H=[\sigma ^*\kappa ]$, and let $\eta $ be the pullback of the Fubini–Study form by the resolved projection to $\mathbb P^{d-1}$. The logarithm defining $U|_ B$ gives

The residual form is smooth semipositive for $a>1$, with class $aH-F$. Its smooth top product off $F$ computes the non-pluripolar mass, and the smooth measure does not charge $F$. Since $H|_ F=0$, all mixed intersections involving both $H$ and $F$ vanish; $H^ d=1$ and $F^ d=(-1)^{d-1}$ give $(aH-F)^ d=a^ d-1$. Formula (20.3.53) now supplies the elliptic factor $2nb$. $\square$

Proof. All sufficiently large base backgrounds give the same divisorial Lelong numbers (20.3.47). On the compact Kähler base, equality of these numbers on all compact models is equivalent to equality of the multiplier ideals at every real positive weight; see the closed criterion recalled in 25.15. Hence all $w_ a$ have the same ideal class. The submersion identity of lemma 5.37.7 transfers the equality to $q^*w_ a$, which does not depend on $b$. Equal ambient multiplier ideals also give equal trace data.

By theorem 20.3.4, the trace data of $U_1+U_2$ is the sum of those of $U_1$ and $U_2$. The sum of two exact base realizers has precisely these divisorial Lelong numbers, so the same closed criterion identifies its ideal class with that of the base trace envelope for $U_1+U_2$. Positive scaling is identical. Pullback gives both assertions on $Y$. $\square$

Proof. For $h\in \mathrm{PSH}(B,a\kappa )$, each finite obstacle envelope $P_{\Omega _{a,b}}(q^*h+A,0)$ is invariant under the elliptic action. Its full supremum descends to the base, where (20.3.50) gives the background condition. Surjectivity makes the obstacle inequalities equivalent on $Y$ and $B$. Hence

Letting $A\to +\infty $ and taking upper semicontinuous regularizations proves $P_{\Omega _{a,b}}[q^*h]=q^*P_{a\kappa }[h]$. Analytic type follows from proposition 18.4.8.

The base models decrease to $w_ a$ by theorem 20.3.27. Their pullbacks therefore decrease to $q^*w_ a$ and converge in $L^1(Y)$ by genuine decreasing quasi-plurisubharmonic convergence. The extracted-divisor formula (20.3.45) transfers convergence of all base divisorial Lelong numbers to every $D$ over $Y$; horizontal divisors have coefficient zero throughout. Finally, lemma 5.43.19 gives $L^1$ convergence on each compact modification. $\square$

Proof. The curvature current of $a$ is the sum of the pullbacks of the two positive currents $A\eta +\mathrm{dd}^{\mathrm{c}}w$ and $\omega _ Z+\mathrm{dd}^{\mathrm{c}}h$, which proves positivity.

Suppose first that both potentials are bounded. Expanding the bounded product by multilinearity, only the terms with one factor from $C$ and $k$ factors from $Z$ survive: two factors from $C$ vanish for dimension reasons, and so do $k+1$ factors from $Z$. The binomial coefficient is $k+1$, which gives the formula for bounded potentials once the exterior product $q^*(A\eta +\mathrm{dd}^{\mathrm{c}}w)\wedge r^*(\omega _ Z+\mathrm{dd}^{\mathrm{c}}h)^ k$ is identified with the product measure $(A\eta +\mathrm{dd}^{\mathrm{c}}w)\otimes (\omega _ Z+\mathrm{dd}^{\mathrm{c}}h)^ k$. To justify this identification at the present regularity, add local plurisubharmonic corrections on each factor and regularize by decreasing convolution on smaller product charts. For the smooth approximants the identity is the exterior product of forms. Local Bedford–Taylor convergence, together with the finite expansion in the smooth background, gives weak convergence of the factor measures, with uniformly bounded variation on compact subcharts. The product measures then converge weakly: one first tests against products of continuous functions on the two factors and then uses uniform approximation by finite sums of such products. This proves the identity in the limit.

For general potentials, subtract constants so that $w,h\leq 0$, and set $w_ j=\max (w,-j)$ and $h_ j=\max (h,-j)$. On

the sum agrees with the bounded potential $w_ j\circ q+h_ j\circ r$. This set is plurifine open and Borel, and it lies in $\{ a>-2j\} $. Locality of the products of bounded truncations therefore identifies the restriction to $R_ j$ of the extended measure of $a$ with that of the bounded sum. By the bounded case, this is $k+1$ times the product of the restrictions of the truncated measures, which are exactly the restrictions of the two extended measures of the factors.

The sets $R_ j$ increase to the product of the complements of the two pole sets, and both sides of (20.3.60) give zero mass to the remaining set: for the left-hand side it is the pole set of $a$, and for the right-hand side this follows because each factor measure charges no pluripolar set and the product measures are $\sigma $-finite. Monotone convergence of positive measures proves the formula, even when the total mass is infinite. $\square$

Proof. Choose the nonpositive potential $H$ of theorem 13.14.18 on $Z$; it has infinite extended mass, and $\mathrm{e}^{-tH}$ is integrable for every $t>0$. Put $a=w\circ q+H\circ r$. The product formula and $A-\delta >0$ show that the mass of $a$ is infinite, while (20.3.61) gives the finite positive mass of $v$.

We prove equality of the multiplier ideals locally, for every $t>0$. Since $H\leq 0$, integrability against $\mathrm{e}^{-ta}$ implies integrability against $\mathrm{e}^{-tv}$, so $\mathcal I(ta)\subseteq \mathcal I(tv)$. Conversely, lemma 5.37.7 gives $\mathcal I(tv)=\mathcal I(tw)\mathcal O_ Y$. Local generators pulled back from $C$ are square integrable against $\mathrm{e}^{-tw}$, so by Fubini's theorem they are also square integrable against $\mathrm{e}^{-tw}\mathrm{e}^{-tH}=\mathrm{e}^{-ta}$; multiplication by bounded holomorphic functions and finite sums preserve this local integrability. Thus $\mathcal I(tv)\subseteq \mathcal I(ta)$, which proves equality at every positive weight. Theorem 5.37.3 then gives equality of all divisorial Lelong numbers after modification.

Finally, since $a$ has infinite mass, proposition 13.14.16 shows that $Y$ does not have the bounded mass property. $\square$

Proof. For $t\leq 0$ put $\chi (t)=1-(1-t)^\alpha $. Then

For $L>0$ set $\epsilon _ L=\chi '(-L)$, $C_ L=-\epsilon _ LL-\chi (-L)$, and define

The values and first derivatives match, and $\chi _ L$ is convex and increasing with derivative at most $\epsilon _ L$. Off $p$, the chain rule gives $\mathrm{dd}^{\mathrm{c}}\chi _ L(G_ p)\geq -\epsilon _ Lw$. Canonical extension across $p$ retains this bound by the removable-singularity theorem, so $U_ L=\chi _ L(G_ p)$ has the required curvature. Near $p$ it differs from $U$ by $C_ L$, and elsewhere both are bounded. Its growth is sublogarithmic, hence its Lelong number is zero.

Writing $x=1+L$, we have $C_ L=(1-\alpha )x^\alpha +\alpha x^{\alpha -1}-1\geq 0$. Also $\chi (t)\leq \epsilon _ Lt$ for $-L\leq t\leq 0$, by integrating $\chi '\geq \epsilon _ L$ on this interval. It follows that $U\leq U_ L\leq 0$. Since $\epsilon _ L\to 0$, the functions tend to zero off $p$, and dominated convergence gives the $L^1$ limit. $\square$

Proof. We first remove the logarithmic divisor. If $v\sim r$, then $\nu (v,D)=b$, so lemma 5.38.1 gives $\theta +\mathrm{dd}^{\mathrm{c}}v-b[D]\geq 0$. Taking $q=v-bG_ p$ in $L^1$ and then its quasi-plurisubharmonic representative gives

Indeed the ordinary inequalities hold off $D$ and extend canonically across it. Conversely such a $q$ gives $v=bG_ p+q$ with

Their locally bounded products agree off $D$, and both extended measures give $D$ zero mass. Thus their extended measures agree everywhere.

If $k>0$, choose $L$ with $\epsilon _ L<k/2$. The potential $bG_ p+U_ L$ belongs to the specified type and satisfies

For necessity, average $q$ over the unitary group acting transitively on the second projective line, with its Haar probability measure. The background is invariant. The average has a genuine first-factor representative $q_0\sim U$ and $kw+\mathrm{dd}^{\mathrm{c}}q_0\geq 0$. One can first average in $L^1$ and then take the canonical representative; the curvature inequality and the ordinary bounds pass to it. The positive curve measure has total mass $k$, excluding $k<0$. When $k=0$, $q_0$ is harmonic on the compact curve and therefore constant, contradicting $q_0\sim U$.

It remains to compute the mass for every competitor, before using the average. Choose a closed Kähler form $K=C(w_1+w_2)\geq \beta $ and a fixed $u_0=U_ L$ as above. The closed positive current $S_0=K+\mathrm{dd}^{\mathrm{c}}u_0$ has no coefficient mass on $D$, since its first-factor curvature has no atom at $p$. The product formula gives

For $S=K+\mathrm{dd}^{\mathrm{c}}q$, both $S$ and $S_0$ are closed positive currents in the same Kähler class, with bounded difference of potentials and small unbounded locus $D$. The classical closed-class comparison of non-pluripolar products, applied in both singularity orders in degrees one and two, gives

Here the degree-one non-pluripolar current is the restriction of $S$ off $D$ and is closed. Hence the positive current $S-\langle S\rangle $ has zero trace against $K$, and vanishes. This proves $\langle S\rangle =S$. Equivalently, absence of polar coefficient mass follows from proposition 5.46.3. The closed comparison is used only for $K$.

Put $R=\beta -K$. Bedford–Taylor expansion off $D$, followed by extension by zero, now yields

The first term is finite by closed comparison, the second has finite variation because $S$ has finite trace, and the last is smooth. Since none has coefficient mass on $D$, this is also an identity on $Y$. Distributional Stokes therefore gives

The pairing uses only $q\in L^1$. Averaging preserves it, since $\mathrm{dd}^{\mathrm{c}}\beta =\mathrm{dd}^{\mathrm{c}}a\wedge w_2$. The last measure is positive of total mass $k$, proving the bounds in (20.3.68) for all original competitors. Substituting $q=v-bG_ p$ and $\int G_ p\mathrm{dd}^{\mathrm{c}}a=a(p)-\int aw$ proves (20.3.67).

To approach either bound, choose a point $x$ where $a$ takes the corresponding extreme value, smooth strictly positive measures $\zeta _ j$ of mass $k$ tending weakly to $k\delta _ x$, and smooth $g_ j$ with $\mathrm{dd}^{\mathrm{c}}g_ j=\zeta _ j-kw$. Choose $L_ j$ so that $\epsilon _{L_ j}$ is less than half the minimum density of $\zeta _ j$ relative to $w$ and $|\int U_{L_ j}\mathrm{dd}^{\mathrm{c}}a|<1/j$. Then

Their masses tend to $2ka(x)$ by (20.3.69). $\square$

Proof. Near $D_ X$, choose a transverse coordinate $z_1$. A germ whose exact transverse order is $j$ has a nonzero leading tangential coefficient. With $t=-\log |z_1|^2$, its radial integrability against $\mathrm e^{-su_ X}$ is determined by

Bounded errors in this exponent do not change convergence. It is integrable exactly when $j+1>sb$, and diverges at equality. Local holomorphic bounds give sufficiency, and a region where the leading tangential coefficient is nonzero gives necessity by Fubini. This proves the ambient ideal formula in the book's convention $|f|^2\mathrm e^{-su_ X}$. Restriction and pullback of this principal ideal give order $\lfloor sb\rfloor \operatorname {ord}_ E(\pi ^*D)$. Divide by $s$ and use theorem 20.3.2.

The residual $U_ L$ has trivial multiplier ideals at every weight, but is unbounded. At a generic point of $D$, an analytic type with divisorial coefficient $b$ is $b\log |z_1|^2+O(1)$. Our residual tends to $-\infty $ like $-t^\alpha $, excluding ordinary equivalence. This also covers $b=0$. $\square$

Proof. By proposition 5.43.18, every $\pi ^*\theta $-plurisubharmonic potential descends. Since $\pi $ is surjective, ordinary inequalities are equivalent before and after pullback. Thus the competitors are in bijection. On the locus where $\pi $ is an isomorphism, locality identifies their products. Both extended products give the excluded analytic subsets zero mass, so

Taking the infimum and supremum proves the identities. Smooth gauge preserves the actual currents and measures; it preserves multiplier ideals because $\mathrm e^{sf}$ is locally bounded above and below. $\square$

Proof. Put $\beta =(c-b)w_1+aw_2$. The representatives of lemma 20.3.65 show that $U$ satisfies (18.27.1); by lemma 18.27.2 so does $\pi ^*U$. The logarithmic divisor is $b\pi ^*D$, whose smooth curvature is $b\pi ^*w_1$, so its residual background is $\pi ^*\beta $. If a strict current existed in the ordinary type of $\pi ^*r$, theorem 18.27.4 would give a smooth $f$ with $\pi ^*\beta +\mathrm{dd}^{\mathrm{c}}f>0$. The exceptional curve $E$ of the last blowup is contracted by the composite map. Restriction to this smooth curve would give $\mathrm{dd}^{\mathrm{c}}(f|_ E)>0$, contrary to its zero integral by Stokes. Only the smooth function $f$ is restricted, so this argument applies also when the original potential has a pole along $E$.

For the one-sided assertion, start with the strict representative $v_0=bG_ p+U_ L$ from theorem 20.3.66. Suppose $v_ j$ is strict with lower bound $\delta _ jh_ j>0$ on the current surface, and let $\pi _{j+1}$ be the next point blowup. For its exceptional divisor $E_{j+1}$, choose a smooth metric on $\mathcal O(E_{j+1})$ with curvature $\eta _{j+1}$ negative along the vertical tangent, and write $\mathrm{dd}^{\mathrm{c}}\ell _{j+1}=[E_{j+1}]-\eta _{j+1}$. For all sufficiently small $t_ j>0$, $\pi _{j+1}^*h_ j-t_ j\eta _{j+1}>0$. To see this, in a blowup chart $(x,z)\mapsto (x,xz)$ the pullback has a uniformly positive horizontal coefficient, mixed coefficient $O(|x|)$ and Schur complement at least $c_1|x|^2$. The perturbation adds a positive vertical term at least $t_ jc_2$, while its error in the Schur complement is bounded by $C(t_ j|x|+t_ j|x|^2+t_ j^2)$. Shrink the exceptional neighborhood and then choose $t_ j$ small; on its compact complement positivity persists under small perturbations. Thus

is strict. These genuine quasi-plurisubharmonic pullbacks are defined even over pole centers. The logarithmic terms are bounded above, so induction gives $v'\leq \pi ^*r+C$. Finally proposition 20.3.73 preserves both volumes in the pulled-back ordinary type. The additional exceptional poles of the one-sided witness are not competitors in that fixed type. $\square$