Proposition 13.24.820027E32C14E

Proof. Ideal profile and envelope. Choose an integer $C>A$ with $C\kappa \geq \theta $. Both $u$ and $v$ are $C\kappa $-plurisubharmonic. The projective valuative characterization of all-weight ideal types applies in this closed background: generic orders on all smooth projective modifications recover point Lelong numbers by further point blowups, and equality of those numbers is equivalent to ideal equivalence. The stated profile therefore implies $v\sim _{\mathcal I}u$. Conversely, ideal equivalence pulls back under every compact modification by lemma 5.43.23; the local valuative criterion theorem 5.37.3 then gives equality of generic coefficients on every such modification. The common factor between growth against $\log |z|$ and the book's $\log |z|^2$ convention does not affect these equalities.

The potential $u$ has complex Hessian $S-A\kappa $. By corollary 18.4.9, its ordinary and ideal envelopes agree and are ordinarily equivalent to $u$. Ideal equivalence identifies the defining families for the two ideal envelopes in the statement, proving the assertion about $q$. The inequality $\theta +\mathrm{dd}^{\mathrm{c}}u\geq R>0$ proves bigness. For any Hermitian form $h$ on $X$, choose $D$ with $D\kappa \geq h$. Bounded $h$-plurisubharmonic potentials are $D\kappa $-plurisubharmonic, and positivity of bounded mixed products bounds their masses by $\int _ X(D\kappa )^2=D^2$. This proves the bounded mass property.

The bounded residual. Fix $w\sim u$ in $\mathrm{PSH}(X,\theta )$ and put $T=C\kappa +\mathrm{dd}^{\mathrm{c}}w$. Its base divisorial coefficients are $a_ j$, so divisorial removal and passage through increasing finite sums give

The distributional difference has a canonical quasi-plurisubharmonic representative, which is bounded by ordinary equivalence and almost-everywhere order. Thus $K$ has bounded local potentials, and its trace charges no pluripolar set. The current $S$ is carried by the actual pole union $\bigcup _ jH_ j\subset \{ w=-\infty \} $. Thus $\langle T\rangle =K$. For every smooth real $(1,1)$-form $L$ with $\mathrm{dd}^{\mathrm{c}}L=0$, distributional Stokes gives

Closed top mass and smooth correction. Put $T_ u=C\kappa +\mathrm{dd}^{\mathrm{c}}u=S+(C-A)\kappa $. For the finite partial sums $u_ N=\sum _{j\leq N}a_ j\ell _ j$, the closed non-pluripolar mass is $(C-\sum _{j\leq N}a_ j)^2$: outside the finite divisor union the current is the smooth residual, and the measure gives that union zero mass. Since $u_0\leq u_ N$, closed-background mass monotonicity, the closed case of theorem 13.17.1, gives

For the reverse inequality, locally write $S=\mathrm{dd}^{\mathrm{c}}b$ with $b$ plurisubharmonic. The currents $(C-A)\kappa +\mathrm{dd}^{\mathrm{c}}\max (b,-k)$ have bounded local potentials and dominate $(C-A)\kappa $. Their squared products, restricted to $\{ b>-k\} $, therefore dominate $(C-A)^2\kappa ^2$ on that set. Letting $k$ increase and using the local truncation definition gives $\langle T_ u^2\rangle \geq (C-A)^2\kappa ^2$. The two bounds prove equality of total masses. Closed mass invariance under bounded changes then gives

Set $L=C\kappa -\theta \geq 0$, so $\mathrm{dd}^{\mathrm{c}}L=0$. Expanding the bounded products of $\max (w,-k)$ in the two backgrounds and restricting to $\{ w>-k\} $ gives, in the limit,

All correction measures are finite: the closed top measure has the mass just computed, the degree-one limit is $K$, and the smooth measure $L^2$ gives the pole set zero mass. The same truncations are used in both backgrounds, so this identity is an equality of Borel measures. Integrating yields

This holds throughout the ordinary type, proving both singularity-volume identities and their strict positivity. $\square$