Pome

Corollary 13.21.4D1480FEAFF87

Corollary 13.21.4.D1480FEAFF87 For every $u\in \mathrm{PSH}(X,\theta )$,

\[ \underline{\operatorname {vol}}_{\theta _ j}(u)\searrow \underline{\operatorname {vol}}_\theta (u). \]

If this limiting volume is positive, then also

\[ \overline{\operatorname {vol}}_{\theta _ j}(u)\searrow \overline{\operatorname {vol}}_\theta (u),\qquad P_{\theta _ j}[u]\searrow P_\theta [u]. \]

Proof. Fix a Hermitian form $\omega $. Uniform convergence gives $\theta _ j\leq \theta +\varepsilon _ j\omega $ with $\varepsilon _ j\to 0$, so the volume limits follow from the two estimates in proposition 13.20.6; monotonicity makes them decreasing. In the positive-lower-volume case, apply proposition 13.21.3 to the constant sequence $u_ j=u$. $\square$

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