18.22 Models over a curve00A545D724DB

Proof. By 5.43.7, every potential descends, even for this possibly indefinite background. Thus nonemptiness is equivalent to the curve condition in lemma 5.29.22. Descent preserves the ordinary order, and 5.43.9 preserves and reflects the all-weight ideal order. At each $x\in X$, choose coordinates centered at $x$ and $p(x)$. A complex line detects the first nonzero homogeneous term of $p$. Its restriction to that line is open, so $p$ is open. Upper regularization therefore commutes with pullback, proving the envelope identities and the model bijections.

Write $\beta =A\eta +\mathrm{dd}^{\mathrm{c}}h$ with $h$ smooth, using lemma 5.29.22. For $v\in \mathrm{PSH}(C,\beta )$, put $w=v+h\in \mathrm{PSH}(C,A\eta )$. Then

When $A>0$, put $W=w\circ p$. Its bounded truncations are $\max (W,-j)=\max (w,-j)\circ p$. Locally choose a smooth $a$ with $\mathrm{dd}^{\mathrm{c}}a=A\eta $. The bounded subharmonic function $a+\max (w,-j)$ has smooth decreasing local approximants. Their pullbacks have complex Hessian rank at most one, including at critical points, so their $n$th powers vanish. Bounded product continuity makes each truncation have zero top product, and hence $W$ has zero extended measure in the positive background. The current invariance in theorem 13.14.3 gives zero extended measure in the original decomposition as well.

When $A=0$, the curve lemma gives $v=-h+c$. Its pullback is smooth with zero curvature current, so its top product is zero. Surjectivity gives the stated normalization. Every potential differs from this one by a constant, and both envelopes fix its normalized representative. Since descent applies to every representative of every ordinary type, both singularity volumes are zero in all cases. $\square$

Proof. The positive $(0,0)$-current $S=p_*(\gamma ^{n-1})$ on $C$ satisfies $\mathrm{dd}^{\mathrm{c}}S=p_*\mathrm{dd}^{\mathrm{c}}(\gamma ^{n-1})=0$. A harmonic distribution is smooth, so $S$ is a constant $\kappa _\gamma $. On the complement of the critical values, smooth fiber integration identifies it with the positive integral of $\gamma ^{n-1}$ on a regular fiber. Thus $\kappa _\gamma >0$.

First let $\mu $ be positive with mass $M>0$. Choose smooth positive measures $\mu _ j$ with the same mass and $\mu _ j\rightharpoonup \mu $. Their mean-zero solutions $q_ j$ have a common lower curvature bound $-M\eta $. Normalized compactness bounds $\sup _ Cq_ j$: their mean-zero condition makes this supremum the negative mean of their supremum-zero normalization. Hence $(q_ j)$ is precompact in $L^1(C)$. Every limit has mean zero and Hessian $\mu -M\eta $. Two such limits differ by a harmonic function, hence by a constant, which their means force to be zero. Therefore $q_ j\to q_\mu $ in $L^1(C)$.

By 5.43.10, $q_ j\circ p\to q_\mu \circ p$ in $L^1(X)$. Thus $p^*\mu _ j\to p^*\mu $ as currents and their smooth Hermitian traces converge weakly. For smooth $\mu _ j$, the definition of $S$ gives $p_*L_\gamma (\mu _ j)=\kappa _\gamma \mu _ j$. Passing to the limit proves the identity for $\mu $. The case $M=0$ is zero, and linearity proves the signed identity.

For a positive $\mu $, split it into $\mathbf1_ B\mu $ and $\mathbf1_{C\setminus B}\mu $. Their traces are positive; the pushforward identity shows that they are respectively carried by $p^{-1}(B)$ and its complement. Additivity then gives the restriction identity. Apply the same argument to the positive and negative Jordan parts of a signed measure. Their disjoint Borel carriers pull back to disjoint carriers, so their traces are the Jordan parts upstairs. The pushforward identity gives the asserted total-variation norm.

Finally the Poincaré–Lelong formula in the book normalization gives $p^*\delta _ s=[F_ s]$, including every zero multiplicity of a local equation of the fiber. Its trace has total mass $\kappa _\gamma $, by the pushforward identity with $\mu =\delta _ s$. $\square$

Proof. The definition of pullback measure gives $p^*T_ v=\theta +\mathrm{dd}^{\mathrm{c}}(v\circ p)$: in the smooth decomposition $\beta =A\eta +\mathrm{dd}^{\mathrm{c}}h$, both sides are $A p^*\eta +\mathrm{dd}^{\mathrm{c}}((v+h)\circ p)$. Apply the Borel restriction and total-variation identities of 18.22.2 to get

The envelope bijection and proposition 13.29.1 give the isometry, completeness and geodesics. Polar measures have mass at most $A$, so their mutual distance is at most $2A$. For $A>0$, the models of $A\delta _ s$ and $A\delta _ t$, $s\ne t$, attain this bound. For $A=0$, 18.22.1 gives the singleton.

Ideal projection corresponds to the atomic part of the polar measure by the same curve classification. If $\mu =\mu _{\mathrm{at}}+\mu _{\mathrm{diff}}$ and $\nu $ is atomic, the measures $\mu _{\mathrm{diff}}$ and $\mu _{\mathrm{at}}-\nu $ are mutually singular. Therefore

The minimum is attained exactly at $\nu =\mu _{\mathrm{at}}$. Taking the atomic part does not increase total variation, giving the Lipschitz estimate. Atomic polar measures of mass at most $A$ form a closed convex subset of the polar-measure space.

For $A>0$, the smooth-gauge model bijection in the proof of proposition 13.29.1 preserves analytic singularities in both directions: its envelope differs by a bounded function from the smoothly transported analytic model, by proposition 18.5.1. Thus the analytic density in corollary 13.28.7 carries from $A\eta $ to $\beta $. The analytic descent equivalence of 5.43.11 and the envelope bijection then give the stated closure upstairs. At zero area the unique smooth model is already analytic and ideal.

Curve metric convergence implies $L^1(C)$ convergence by proposition 13.29.1, and 5.43.10 gives convergence upstairs. Finally all top-degree masses, including those of pairwise maxima, vanish by 18.22.1. The upper-volume comparison therefore vanishes. $\square$

Proof. Put $r=P_\beta [v]_{\mathcal I}$. The envelope bijection gives $R=r\circ p$. By corollary 13.29.2, base envelope equality is equivalent to atomic polar curvature and to $v\in \mathcal E_{\mathrm{env}}(C,\beta ,r)$. For each stretching parameter $b\geq 1$, descent and surjectivity identify competitors satisfying $b^{-1}h+(1-b^{-1})r\leq v$ with their upstairs counterparts. This uses only convex combinations and remains valid at poles. Thus the relative-class conditions correspond. Apply proposition 18.12.2 for equality below every smooth obstacle on $X$; the converse includes the zero obstacle. The zero-area assertion follows from the smooth singleton. $\square$