PomeFB4254C93740FB4254C93740
Let $Y$ be a connected compact complex manifold of dimension $n\geq 1$, let $\theta $ be a smooth real $(1,1)$-form on $Y$, and let $\pi :Z\to Y$ denote a smooth compact modification.
When the background $\theta $ is not closed, subtracting a closed divisorial current from its pullback leaves the exterior derivative unchanged, so the residual currents of section 20.4 are not closed and define no Bott–Chern class. They do define classes in an affine space over Bott–Chern cohomology, since two representatives with the same exterior derivative differ by a closed current. The trace envelope still gives positive residual representatives, and the exceptional divisors of subsection 20.4.25 describe how the classes change between models.
Analytic singularities give nef Cartier realizations in these affine spaces. For a general potential, analytic regularization gives such realizations in slightly larger backgrounds, and on each fixed model the positive residual currents converge weakly. The error in the background tends to zero smoothly, but removing it can destroy nefness. Nor does this linear convergence control Monge–Ampère mass: for a Hermitian background with unbounded upper mass, a potential of infinite mass can have the same residual classes as the zero potential, whose classes are realized by a smooth positive form.
89D86926636ATwo real $(1,1)$-currents on $Z$ with exterior derivative $\pi ^*\mathrm{d}\theta $ differ by a closed current, and modulo $\mathrm{dd}^{\mathrm{c}}$ of global distributions such differences are Bott–Chern classes. Let $\mathcal D'(Z,\mathbb R)$ denote the space of real distributions on $Z$, and set
Write $\{ T\} $ for the class of $T$ in this space. By lemma 20.4.10, every class has a smooth representative, and translation by Bott–Chern classes gives
A smooth real $(1,1)$-form $\eta $ represents an element of $\mathscr A_ Z(\theta )$ exactly when $\mathrm{d}\eta =\pi ^*\mathrm{d}\theta $. As for Bott–Chern classes, pullback of a class is defined on a smooth representative and pushforward on current representatives.
Proposition 10.30.2.5F6B6BF8B562 If $g:Z'\to Z$ is a compact modification over $Y$, pushforward and pullback induce maps
Pushforward preserves classes with positive current representatives.
Proof. Write $\pi '=\pi \circ g$. Translation by the pulled-back background identifies the required maps with the Bott–Chern maps of subsection 20.4.9. Indeed, for a closed current $C'$ on $Z'$ and a smooth closed form $\beta $ on $Z$,
where the first equality uses the change of variables formula for the degree-one map $g$ on the smooth background. Since the Bott–Chern maps are well defined on the closed terms, these identities define the affine maps independently of the representatives. Their composition is the identity because $g_*g^*$ is the identity both on the background and on Bott–Chern classes. Finally, pushforward under a proper map preserves positive currents. $\square$
The prescribed derivative is essential for this pushforward: $g_*\pi '^*\mathrm{d}\theta =\pi ^*\mathrm{d}\theta $ is smooth, whereas the pushforward of an arbitrary smooth exact form can be a singular current.
We call a family $(\mathbb A_ Z)$, with $\mathbb A_ Z\in \mathscr A_ Z(\theta )$ for every model $Z$ and compatible with pushforward, a b-divisor class over the background $\theta $. Pseudoeffectivity and Cartier realizations are defined as in definition 20.4.11, with affine classes in place of Bott–Chern classes. A class in $\mathscr A_ Z(\theta )$ is nef if, for a Hermitian form $\gamma $ on $Z$, it has for every $\epsilon >0$ a smooth representative bounded below by $-\epsilon \gamma $, as in subsection 20.4.16.
C43F5C12FA6FSince the trace envelope dominates the trace currents on every model, subtracting them from its curvature current gives positive residual representatives, with exterior derivative $\pi ^*\mathrm{d}\theta $. The classes are determined by the prescribed trace currents, even when the envelope has larger Lelong numbers.
In this section, $Y$ is a complex submanifold of a connected compact complex manifold $X$. Let $\Theta $ be a smooth real $(1,1)$-form on $X$ with $\Theta |_ Y=\theta $, and let $u\in \mathrm{PSH}(X,\Theta )$ have generic Lelong number $\nu (u,Y)=0$. We use the trace currents $S_ Z(u;Y)$ of subsection 20.4.1, now with $\Theta $ in the role of the ambient background.
Theorem 10.30.4.28C8F709D1CC The family defined by
is a pseudoeffective b-divisor class over $\theta $. It is unchanged by ambient multiplier-ideal equivalence and, after the resulting affine spaces are identified, by a smooth $\mathrm{dd}^{\mathrm{c}}$ change of the restricted background.
Proof. The currents $S_ Z$ are closed, so the classes belong to the indicated affine spaces, and their compatibility follows from theorem 20.4.5, proposition 10.30.2. Take $v=\mathcal T_\theta (u;Y)$, the trace envelope of (20.3.25). By proposition 20.4.8,
on every model, which proves pseudoeffectivity in the original restricted background. The currents $S_ Z$ depend only on the ambient multiplier ideals. For a smooth real function $f$ on $Y$, the change $\theta \mapsto \theta +\mathrm{dd}^{\mathrm{c}}f$ adds $\mathrm{dd}^{\mathrm{c}}(f\circ \pi )$ to every representative, which is zero in the quotient. This proves both invariances. $\square$
When $\theta $ is closed, this family is the Bott–Chern family of proposition 20.4.12.
On a higher model, the trace current can exceed the pullback of the trace current from a lower model. The effective exceptional difference of theorem 20.4.26 therefore translates the residual class, and the translation preserves the prescribed exterior derivative.
Corollary 10.30.5.9157A5A52988 Let $g:Z'\to Z$ be a compact modification over $Y$, and let $E_ g$ be the effective exceptional divisor of theorem 20.4.26. The trace classes of theorem 10.30.4 satisfy
where the Bott–Chern class on the right acts by translation on the affine space $\mathscr A_{Z'}(\theta )$.
Proof. Let $\pi :Z\to Y$ be the structure map and put $\pi '=\pi \circ g$. Abbreviate $S_ Z=S_ Z(u;Y)$, and use lemma 5.38.4 to write $S_ Z=B_ Z+\mathrm{dd}^{\mathrm{c}}v_ Z$ with $B_ Z$ smooth and closed and $v_ Z$ quasi-plurisubharmonic. The affine class $\mathbb B_ Z$ has the smooth representative $\pi ^*\theta -B_ Z$. Its pullback is therefore the class of $\pi '^*\theta -g^*S_ Z$, since the two representatives differ by $\mathrm{dd}^{\mathrm{c}}(v_ Z\circ g)$. Substituting $S_{Z'}=g^*S_ Z+[E_ g]$ gives the asserted identity. Both representatives have exterior derivative $\pi '^*\mathrm{d}\theta $, because the subtracted currents are closed, so the subtraction takes place within $\mathscr A_{Z'}(\theta )$. $\square$
The cocycle identity (20.4.28) makes these translations compatible with composition of modifications, and by theorem 20.4.26 their analytic-ideal approximants converge in total variation of the trace measures. On the other hand, theorem 20.3.30 identifies the trace envelopes on the models with pullbacks of the original trace envelope. The exceptional translation accounts for the additional divisorial subtraction on the higher model; it is consistent with this exact pullback of the envelope, and it can be nonzero even when the trace data are realized exactly, as in example 20.4.31.
For the rest of the chapter, the ambient manifold is $Y$ itself, and $u\in \mathrm{PSH}(Y,\theta )$. For a smooth compact modification $\pi :Z\to Y$, we write $N_ Z(u)$ for the divisorial current of $u\circ \pi $ and $\mathbb D_\theta (u)$ for the resulting residual family. By the self-trace identity of lemma 20.4.14,
where $E$ ranges over the prime divisors on $Z$. By theorem 10.30.4, $\mathbb D_\theta (u)$ is pseudoeffective. In this case divisorial removal gives the positive representative $\pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(u\circ \pi )-N_ Z(u)\geq 0$ directly, by lemma 5.38.1 and passage to the limit of increasing finite sums.
AFB10A9AB20BOn a resolution of an analytic singularity, the bounded residual potential has smooth approximants with arbitrarily small curvature error, and these prove nefness within the same affine class. Boundedness also prevents new divisorial terms on further modifications, which gives the Cartier realization.
Proposition 10.30.7.A8BE1F55208F If $u\in \mathrm{PSH}(Y,\theta )$ has analytic singularities, then $\mathbb D_\theta (u)$ has a nef Cartier realization on a smooth compact modification.
Proof. Resolve the analytic singularity ideal as in the proof of theorem 20.4.18. On a suitable model $\pi :Z\to Y$ we obtain
where $D$ is a finite effective rational divisor, $\beta $ is smooth with $\mathrm{d}\beta =\pi ^*\mathrm{d}\theta $, and $v$ is globally bounded after the choice of metrics on the line bundles of the components of $D$. Positivity of $R$ follows from divisorial removal.
Fix a Hermitian form $\gamma $ on $Z$ and apply proposition 5.31.2 to $v$ in the smooth background $\beta $. It gives smooth functions $v_ j$ with $\beta +\mathrm{dd}^{\mathrm{c}}v_ j\geq -\epsilon _ j\gamma $ and $\epsilon _ j\downarrow 0$. These forms represent the same affine class and have exterior derivative $\pi ^*\mathrm{d}\theta $, which proves nefness.
On every further modification $g:Z'\to Z$, the current $g^*\beta +\mathrm{dd}^{\mathrm{c}}(v\circ g)$ is positive by lemma 5.43.2. Its bounded potential has zero Lelong numbers, so the divisorial part of the full pullback is exactly $g^*D$. The residual affine class on $Z'$ is therefore $g^*\{ R\} $, which proves the Cartier realization. $\square$
For general $u$, add twice the curvature error of analytic regularization to the background. This gives positive currents with analytic singularities, to which the proposition applies, while the ideal orders identify the limiting residual family.
Theorem 10.30.8.250F831D6E96 Let $u\in \mathrm{PSH}(Y,\theta )$ and fix a Hermitian form $\omega $ on $Y$. There are potentials $u_ m$ with analytic singularities, converging to $u$ in $L^1(Y)$, and positive numbers $\epsilon _ m\downarrow 0$, indexed by sufficiently large positive integers $m$, such that
For each $m$, the family $\mathbb D_{\theta _ m}(u_ m)$ has a nef Cartier realization. On every fixed compact model $\pi :Z\to Y$, its class has the current representative
where $E$ ranges over the prime divisors on $Z$ and the order is computed after pulling the ideal back to $Z$. These representatives satisfy
and the positive residual representatives satisfy
The exterior derivatives of both $C_{m,Z}$ and $R_{m,Z}$ converge smoothly to $\pi ^*\mathrm{d}\theta $.
Proof. Take $u_ m$ from corollary 5.36.2; they converge to $u$ in $L^1$. If $e_ m\to 0$ are their curvature errors, put $\epsilon _ m=\sup _{\ell \geq m}e_\ell +1/m$. These numbers are positive and decrease to zero, and the indices $m$ are unchanged. Adding $2\epsilon _ m\omega $ to the background turns the lower bound $-\epsilon _ m\omega $ into $\epsilon _ m\omega $. Since each $u_ m$ has analytic type $(1/m,\mathcal I(mu))$, proposition 10.30.7 gives a nef Cartier realization in the background $\theta _ m$.
On a fixed model, the generic divisorial Lelong numbers of $u_ m\circ \pi $ are exactly $m^{-1}\operatorname {ord}_ E\mathcal I(mu)$, so $N_ Z(u_ m)=D_{m,Z}$ and $C_{m,Z}=\pi ^*\theta _ m-N_ Z(u_ m)$ represents the class of $\mathbb D_{\theta _ m}(u_ m)$ on $Z$. The self-trace identity and theorem 20.4.2 give $D_{m,Z}\to N_ Z(u)$ in the trace mass of subsection 20.4.1, hence weakly as currents. Since the smooth error $2\epsilon _ m\pi ^*\omega $ tends to zero with all its derivatives, $C_{m,Z}$ converges as stated.
Divisorial removal gives $R_{m,Z}\geq 0$. All the $u_ m$ and $u$ have a common smooth lower curvature bound: choose $A>0$ with $\theta +\epsilon _ m\omega \leq A\omega $ for every index $m$ considered. By lemma 5.43.19, the convergence in $L^1(Y)$ implies $u_ m\circ \pi \to u\circ \pi $ in $L^1(Z)$. Applying $\mathrm{dd}^{\mathrm{c}}$ and adding the convergent currents $C_{m,Z}$ proves the convergence of the residual currents, and positivity persists in the weak limit. Finally, since each $D_{m,Z}$ is closed,
which proves the smooth convergence of the exterior derivatives. $\square$
The classes $\mathbb D_{\theta _ m}(u_ m)_ Z$ lie in $\mathscr A_ Z(\theta _ m)$. Translation by $-2\epsilon _ m\pi ^*\omega $ identifies this space with $\mathscr A_ Z(\theta )$, but subtracting this form need not preserve nefness. Thus the convergence of currents above does not give an approximation by nef Cartier classes with exactly the prescribed exterior derivative.
47AA4DD512F8Now let the background be a Hermitian form $\omega $ on $Y$, with upper mass $v_+(\omega )$ (definition 13.14.14) and extended Monge–Ampère measure $\mathrm{MA}^{\mathrm{ext}}_\omega $ (theorem 13.14.3). When $v_+(\omega )=+\infty $, the potential of theorem 13.14.18 has the same multiplier ideals as a smooth potential. The two potentials therefore have the same residual classes on every model, while their extended masses differ.
Proposition 10.30.10.6436A9863EBA If $v_+(\omega )=+\infty $, there is $U\in \mathrm{PSH}(Y,\omega )$ such that
The common family has the nef Cartier realization $(Y,\{ \omega \} )$.
Proof. Take $U$ from theorem 13.14.18. All its multiplier ideals at positive weights are trivial, so every divisorial order in $D_{m,Z}$ vanishes on every model. The self-trace identity gives $N_ Z(U)=0$, which is also the divisorial part of the zero potential. Both families therefore consist of the pullbacks of the class of $\omega $. Since $\omega $ is smooth and positive, its affine class is nef and realizes the family on $Y$ itself. The extended mass of $U$ is infinite by construction, whereas the smooth measure $\omega ^ n$ has finite mass because $Y$ is compact. $\square$
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