16.29 Envelope systems over Hermitian perturbations4AD4DDD929EE

Proof. By corollary 18.4.4, a $P_\theta [\cdot ]_{\mathcal I}$-fixed potential is also fixed by the ordinary envelope $P_\theta $. The regularized supremum of the curve is $\Gamma _ a$, since the curve is decreasing and constant for $\tau \leq a$. Hence all the conditions of definition 17.17.2 hold, and theorem 17.17.5 applies. The term $\tau =a$ in the transform gives the lower bound, and the inequalities $\Gamma _\tau \leq \varphi $ for $\tau <c$ give the upper bound. $\square$

Proof. Since $\beta \geq \theta $, every slice is $\beta $-plurisubharmonic, and by theorem 18.4.2 its projection is a $\beta $-plurisubharmonic function fixed by $P_\beta [\cdot ]_{\mathcal I}$. Decreasingness, constancy for $\tau \leq a$, and order preservation follow from proposition 18.4.6.

For concavity, let $r,t<c$ and $0<s<1$, and put $\tau =(1-s)r+s t$. By the same proposition and the concavity of the original curve,

Here the first inequality follows by combining the scaling and addition formulas of that proposition, with backgrounds $(1-s)\beta $ and $s\beta $.

Each projection preserves the multiplier ideals at every weight. Hence the two sides of the transition identity are $P_\beta [\cdot ]_{\mathcal I}$-fixed representatives of the same multiplier-ideal class, and they are equal. If two projected curves agree, their original slices have the same multiplier ideals; since those slices are $P_\theta [\cdot ]_{\mathcal I}$-fixed, they agree as well. This proves injectivity. $\square$

Proof. Step 1: the limit. Fix $\tau <c$ and put $q_ j=\Gamma ^{h/j}_\tau $. By compatibility, $q_ j=P_{\theta +h/j}[q_{j+1}]_{\mathcal I}$, and $q_{j+1}\leq 0$ is a competitor for this envelope; hence $q_ j\geq q_{j+1}$. The functions $q_ j$ have supremum zero and a common smooth lower curvature bound. By theorem 5.25.2, corollary 5.25.3, their decreasing limit $q$ is not identically $-\infty $ and has supremum zero. Passing to the limit in the distributional inequalities

gives $q\in \mathrm{PSH}(X,\theta )$.

The defining family of an ideal envelope grows when the background increases. Together with $q\leq q_ j$ and order preservation, this gives

Taking the infimum over $j$ proves $P_\theta [q]_{\mathcal I}=q$. Concavity, decreasingness and constancy on the left half-line pass to the pointwise limits, so these limits form a curve in $\mathscr T_{a,c}(\theta )$.

Step 2: independence of $h$. For every $\eta >0$ there is $j$ with $h/j\leq \eta $, and compatibility then gives $\Gamma ^\eta _\tau \geq \Gamma ^{h/j}_\tau $. Consequently

which proves that $\mathsf R\boldsymbol \Gamma $ does not depend on $h$. This formula also shows that $\mathsf R$ preserves order.

Step 3: the two compositions. Write $q=(\mathsf R\boldsymbol \Gamma )_\tau $. Since $q\leq \Gamma ^\eta _\tau $ and $\Gamma ^\eta _\tau $ is fixed by its envelope, order preservation gives $P_{\theta +\eta }[q]_{\mathcal I}\leq \Gamma ^\eta _\tau $. This proves $\mathsf E\mathsf R\leq \mathrm{id}$.

Now start with a curve $\Gamma \in \mathscr T_{a,c}(\theta )$. Its slice $\Gamma _\tau $ is a competitor in every larger background, so

The two outer terms have the same multiplier ideals, so the two ideal inclusions given by these inequalities show that the middle term has those ideals as well. Since both the middle term and $\Gamma _\tau $ are $P_\theta [\cdot ]_{\mathcal I}$-fixed, they are equal. This proves $\mathsf R\mathsf E=\mathrm{id}$.

Step 4: the adjunction. By the transition proposition, $\mathsf E$ preserves order. If $\mathsf E(\Gamma )\leq \boldsymbol \Gamma $, applying $\mathsf R$ gives $\Gamma =\mathsf R\mathsf E(\Gamma )\leq \mathsf R\boldsymbol \Gamma $. Conversely, if $\Gamma \leq \mathsf R\boldsymbol \Gamma $, applying $\mathsf E$ and using $\mathsf E\mathsf R\leq \mathrm{id}$ gives $\mathsf E(\Gamma )\leq \boldsymbol \Gamma $. This proves (16.29.10), and the final maximality assertion follows from it. $\square$

Proof. For a realized system, this follows from $\mathsf R\mathsf E=\mathrm{id}$ and the preservation of multiplier ideals by each projection. Conversely, the assumed equivalence and the fact that $\Gamma ^\eta _\tau $ is fixed by its envelope in the larger background give

Thus the system equals $\mathsf E\mathsf R(\boldsymbol \Gamma )$. $\square$

Proof. Subtracting $f$ changes the background from $\theta $ to $\theta '$ and preserves concavity, decreasingness and constancy on the left half-line. Slicewise projection preserves these properties by the argument of proposition 16.29.5. Since $f$ is smooth, every new slice has the multiplier ideals of the original one. Applying the construction with $-f$ therefore gives a $P_\theta [\cdot ]_{\mathcal I}$-fixed slice with the same ideals as the original slice, hence the original slice itself. This proves that $\mathsf G_{-f}$ is the inverse of $\mathsf G_ f$; the same uniqueness of fixed representatives proves the cocycle law.

Compatibility of the transformed system and commutation with $\mathsf E$ follow by comparing multiplier ideals in the common final background. For commutation with $\mathsf R$, the gauge estimate of proposition 18.5.1, applied in each background $\theta +\eta $, gives

with constants independent of $\eta $. Pass to the decreasing limits along a cofinal sequence. The new limit has the same ordinary singularity type as the old limit minus $f$, and hence the same multiplier ideals. It is $P_{\theta '}[\cdot ]_{\mathcal I}$-fixed by theorem 16.29.9, so it equals $P_{\theta '}[(\mathsf R\boldsymbol \Gamma )_\tau -f]_{\mathcal I}$. This proves commutation with $\mathsf R$. $\square$

Proof. Take the torus $X$ and the smooth perturbation of proposition 18.5.2, so that $\omega '=\omega +\mathrm{dd}^{\mathrm{c}}f$ is Hermitian and $\int _ X(\omega ')^2\neq \int _ X\omega ^2$, and fix any Hermitian form $h$. In every background $\omega +\eta $, let every slice with $\tau <c$ be zero. This is an envelope system for every $a<c$, since zero is the multiplier-ideal envelope of the unit-ideal class in a positive background. Its first limit is

The gauge bijection sends every slice to $P_{\omega '+\eta }[-f]_{\mathcal I}=0$, since $f$ is smooth and the new background is positive. The second limit is therefore $\int _ X(\omega ')^2$, which is different. $\square$

Proof. Let $s_ D$ be the canonical section of $\mathcal O_ X(D)$, with zero divisor $D$. For a real parameter $t>0$, define a smooth Hermitian metric on this line bundle by prescribing the norm of $s_ D$:

To check that it extends smoothly, use the coordinate $\xi =1/w$ near $D$. The squared norm of a local nonvanishing frame is

which is smooth and positive. The prescribed norm is invariant under the transformations on the complement of $D$, so these local extensions define a global metric. Its curvature form is

Set $\theta =\theta _1$, and let $\sigma =\sigma _0/\pi $ on $E$, where the denominator is the number $\pi $.

We now estimate the curvature from below. Put $r=|v|^2$ and let $H$ be the complex Hessian of $\log (1+tr)$ in the coordinates $(z,w)$. Direct differentiation gives

Consequently

Completing the square gives $\theta _ t\geq -(t/4)\pi ^*\sigma $ on $X\setminus D$, and by continuity this inequality extends across $D$. The restriction of $\theta _ t$ to every fiber is a positive Fubini–Study form, including at infinity. Thus $\theta _ t+A\pi ^*\sigma $ is strictly positive off $D$ whenever $A>t/4$. At $\xi =0$, its coefficient matrix, with the common factor $\mathrm i/(2\pi )$ omitted, is

Its vertical coefficient is positive and its Schur complement is $A-t/4>0$, so the form is strictly positive along $D$ as well. In particular, $\kappa =\theta +\pi ^*\sigma _0$ is Kähler.

The function

extends smoothly across $D$ and satisfies $\theta _ t=\theta +\mathrm{dd}^{\mathrm{c}}f_ t$. Letting $t\downarrow 0$ in the curvature lower bound shows that the class of $\theta $ is nef in the sense of subsection 20.4.16.

Write $\psi _ D=\log |s_ D|_1^2$. By the Poincaré–Lelong formula, $\theta +\mathrm{dd}^{\mathrm{c}}\psi _ D=[D]$. Let $u\in \mathrm{PSH}(X,\theta )$. Then $u-\psi _ D$ is plurisubharmonic on $X\setminus D$. Its lift $\Phi (z,w)$ to $\mathbb C^2$ is invariant under the two deck transformations and, since $u$ is bounded above on the compact surface $X$, satisfies

for some real constant $C$. For each fixed $w$, the function $z\mapsto \Phi (z,w)$ is either identically $-\infty $ or subharmonic. It is $1$-periodic and has logarithmic upper growth, so lemma 5.24.6 makes it constant. Thus $\Phi (z,w)=\phi (w)$, where $\phi $ is a subharmonic function of $w$, not identically $-\infty $. The second deck transformation gives $\phi (w+1)=\phi (w)$, and the bound at $z=0$ gives logarithmic growth, so the same lemma makes $\phi $ constant. Therefore $u=\psi _ D+C$ off $D$, and lemma 5.24.3 extends this equality across $D$. Hence $\theta +\mathrm{dd}^{\mathrm{c}}u=[D]$. $\square$

Proof. For each Hermitian form $\eta $, the nef approximation of lemma 16.29.16 gives a smooth function $f_\eta $ such that $\theta +\eta +\mathrm{dd}^{\mathrm{c}}f_\eta \geq 0$. Then

Let every slice $\Gamma ^\eta _\tau $, $\tau <c$, of the component $\Gamma ^\eta $ be $q_\eta =P_{\kappa +\eta }[u_\eta ]_{\mathcal I}$, with cutoff $c$. The multiplier ideals of $q_\eta $ are those of $\psi _ F$ and do not depend on the smooth function $f_\eta $. Hence the compatibility condition follows by comparing these ideals in the larger background, and the constant curves form a system with the required parameters.

Let $q$ be its limiting slice under $\mathsf R$, and let $\nu (q,F)$ be its generic Lelong number along $F$. Each $q_\eta $ has generic Lelong number one along $F$, so lemma 5.38.7, applied to a decreasing cofinal sequence, gives $\nu (q,F)\geq 1$. By lemma 5.38.1,

By the Poincaré–Lelong identity for $\psi _ F$, this current equals $\theta +\mathrm{dd}^{\mathrm{c}}(q-\psi _ F)$, where the difference $q-\psi _ F$ is first taken as a locally integrable function. Since the current is positive, this difference has a canonical $\theta $-plurisubharmonic representative, and by lemma 16.29.16 it differs from $\psi _ D$ by a constant. Hence

The divisors $F$ and $D$ are smooth and meet transversely, since one is a fiber and the other a section. A direct integrability calculation in local product coordinates therefore gives the stated ideals for every real $t>0$. In particular, the Lelong number at a generic point of $D$ has increased from zero to one, and proposition 16.29.11 shows that recovery fails.

Finally, since $\mathsf R\mathsf E=\mathrm{id}$, the system $\boldsymbol \Gamma $ and the realized system $\mathsf E\mathsf R\boldsymbol \Gamma \neq \boldsymbol \Gamma $ have the same image under $\mathsf R$. Thus $\mathsf R$ is not injective, and $\boldsymbol \Gamma $ is not in the image of $\mathsf E$. $\square$

Proof. The potential $u_\eta =\psi _ F+f_\eta $ has analytic singularities, so proposition 18.4.8 gives $q_\eta \sim u_\eta \sim \psi _ F$. Removing the divisorial component $[F]$ gives a positive current in the background $\theta +\eta $, whose global potential $w_\eta =q_\eta -\psi _ F$ is bounded by this ordinary equivalence.

Off $F$, plurifine locality identifies the Monge–Ampère measure of $q_\eta $ with $(\theta +\eta +\mathrm{dd}^{\mathrm{c}}w_\eta )^2$. This bounded product gives no mass to $F$ by proposition 5.28.2. For closed $\eta =\epsilon \kappa $, Stokes' theorem gives its total mass as $\int _ X(\theta +\epsilon \kappa )^2$.

The normal bundle of $D$ is trivial. Indeed, the transformation $w\mapsto w+1$ becomes $\xi \mapsto \xi /(1+\xi )$, whose derivative at $\xi =0$ is one, and the other transformation fixes $w$. Hence $\int _ X\theta ^2=D^2=0$, and expanding the square gives the formula. The limiting potential has curvature current $[F]+[D]$, which vanishes away from the pluripolar set $F\cup D$; its non-pluripolar measure is therefore zero. $\square$

Proof. First, an $E$-invariant $q\in \mathrm{PSH}(X,bh)$ descends to an upper semicontinuous function $w$ on $B$, since $\pi $ is proper and $q$ is constant on its fibers. In a product chart, Fubini gives $w\in L^1_{\mathrm{loc}}$, and

Lift a smooth base vector to a vector annihilated by $\partial s$. Its pairing with $\beta $ is zero. Testing the positive current above against these lifts and a nonnegative smooth fiber density of integral one proves $b\kappa +\mathrm{dd}^{\mathrm{c}}w\geq 0$ distributionally. A local potential of $b\kappa $ gives its canonical plurisubharmonic representative. Its pullback agrees almost everywhere with $q$, hence everywhere by lemma 5.24.3. Thus $w\in \mathrm{PSH}(B,b\kappa )$. Conversely, every such $w$ lifts to a $bh$-plurisubharmonic function by the displayed identity.

Now set $q=P_{bh}[\pi ^*v]_{\mathcal I}$. Translation by any $g\in E$ preserves its complete competitor family: it preserves the background, the obstacle zero, and the ideals of the invariant input. The envelope is their greatest member, so $q\circ g=q$. Descend $q=\pi ^*w$ as above. By lemma 5.37.7, its ideal equivalence to $\pi ^*v$ implies $w\sim _{\mathcal I}v$: a germ pulled back from the base is weighted-integrable in a product chart if and only if the base germ is, by Fubini. Hence $w$ is a base competitor. Every base competitor also lifts to an ambient competitor by the same lemma and the current identity. Taking the greatest members gives equality. $\square$

Proof. The descent in proposition 16.29.20 gives $\Gamma _\tau ^{\epsilon h}=\pi ^*\phi _\tau ^\epsilon $ with $\phi _\tau ^\epsilon \in \mathrm{PSH}(B,(1+\epsilon )\kappa )$. Normalization, decrease, concavity and constancy for $\tau \leq a$ descend pointwise. The fiber-envelope identity descends both ideal-model fixedness and scalar compatibility:

In particular, the positive scalar slices have the same ideals at every weight.

For a Hermitian form $\xi >0$ on $B$, choose $\delta >0$ with $\delta \kappa \leq \xi $, fixed for the whole parameter interval, and define

The definition is independent of $\delta $ because the scalar inputs have the same ideals. It reproduces the scalar components, and proposition 16.29.5 proves that each component is a curve in $\mathscr T_{a,c}(\kappa +\xi )$. Projection preserves the ideals, so comparing the complete competitor families proves compatibility for $\xi \geq \xi '$. Thus $\boldsymbol \Psi $ is a full envelope system. Every possible extension is forced by this formula, which proves uniqueness.

The retraction theorem applies to both systems. Taking their decreasing limits along $h/j$ and $\kappa /j$ in the scalar identity gives $\Gamma _\tau =\pi ^*\psi _\tau $. In particular the limits, including their values at poles, agree. For any other Hermitian metric $k$ on $B$, constants $0<l\leq r$ with $l\kappa \leq k\leq r\kappa $ give

Both outer sequences decrease to $\psi _\tau $. This also identifies the transverse limit directly, without assuming that it keeps the ideals of the positive scalar slices. $\square$

Proof. The closed Kähler background $\kappa $ is big and $B$ has the bounded mass property. Apply theorem 13.21.7 to the full transverse system. Its initial-volume hypothesis is exactly the displayed one, and it gives

The fiber-envelope identity yields $\Gamma _\tau ^{\epsilon h}=P_{(1+\epsilon )h}[\pi ^*\psi _\tau ]_{\mathcal I}$. For a general $\eta >0$, choose $\epsilon h\leq \eta $. Compatibility and the nested-background identity of proposition 16.29.5 then give

Proof. The function $\lambda $ is positive, increasing and convex, and $u_0\leq 0$, so $u_\tau $ decreases, is pointwise concave and is constant for $\tau \leq a$. Moreover,

Set $q_\tau =P_\kappa [u_\tau ]_{\mathcal I}$. The order, scaling and addition inequalities of proposition 18.4.6 show that $q_\tau $ remains decreasing and concave; it is constant on the same left half-line. Thus $q\in \mathscr T_{a,c}(\kappa )$. Apply the same argument upstairs to $Q_\tau =P_ h[\pi ^*u_\tau ]_{\mathcal I}$. The fiber-envelope identity gives $Q_\tau =\pi ^*q_\tau $, and the nested-background identity identifies the displayed family with $\mathsf E(Q)$. It is therefore an envelope system, its limit is $Q$ by $\mathsf R\mathsf E=\mathrm{id}$, and its scalar slices are invariant.

Apply 13.24.8 with background $\kappa $, coefficients $a_ j=\lambda (\tau )q_ j$ and residual $(1-\lambda (\tau ))\kappa $. It gives the asserted exact lower and upper volumes of $q_\tau $. The transverse recovery theorem applies, also confirming recovery directly.

For every $j$, the divisorial coefficient of $u_\tau $ along $H_ j$ is positive. Projection preserves all ideals and hence these coefficients. Analytic singularities on a compact manifold have only finitely many divisorial components, since their defining coherent ideals have this property. Thus no analytic singularity type has the same all-weight ideals. The same conclusion holds upstairs along the distinct divisors $\pi ^{-1}(H_ j)$, by submersion pullback. $\square$