18.6.7 Comparison with scalar normalization0F1DD25AE72E

Example 18.6.8.0F685C3DE85F Let $\theta $ be semipositive and big, and let $a\in \mathrm{PSH}(X,\theta )$ have analytic singularities with $a\leq 0$. Use the normalization scalars $B_\varphi $ and the sets $\mathcal A_\varphi (G)$ of supporting probabilities from section 11.33, section 11.34, defined with the fixed volume form $\mathrm{d}V$. Set

By proposition 18.4.8, $\varphi _ t\sim ta$; since the ordinary envelope depends only on the ordinary type, $P_\theta [\varphi _ t]=\varphi _ t$. Moreover $ta\in \mathrm{PSH}(X,t\theta )$ and $ta\leq \varphi _ t$, so the envelope $P_{t\theta }(\varphi _ t)$ in the smaller background is not identically $-\infty $. By proposition 11.39.1,

When $\theta $ is Hermitian and $G$ is fixed and smooth, every choice in $\mathcal A_{\varphi _ t}(G)$ converges weakly in $L^1$ to the smooth Gauduchon adjoint probability, by corollary 11.39.6, theorem 11.34.9.