Lemma 18.22.1717F85615B2A

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. $\square$