Pome

Proposition 13.24.271B65CF5AD10

Proposition 13.24.2.71B65CF5AD10 Let $u\in \mathrm{PSH}(X,\theta )$, put $q=P_\theta [u]_{\mathcal I}$, and suppose that $\underline{\operatorname {vol}}_\theta (q)>0$. Then the following conditions are equivalent:

  1. $P_\theta [u]=P_\theta [u]_{\mathcal I}$;

  2. $\overline{\operatorname {vol}}_\theta (u)=\mathcal V^+_{\mathcal I,\theta }(u)$;

  3. $u\in \mathcal E_{\mathrm{env}}(X,\theta ,q)$;

  4. $P_{\theta ,F}[u]=P_{\theta ,F}[u]_{\mathcal I}$ for every $F\in C^\infty (X,\mathbb R)$.

If they hold, then also $\underline{\operatorname {vol}}_\theta (u)=\mathcal V^-_{\mathcal I,\theta }(u)$.

Proof. Recall that $q$ is an ordinary model and $u\preceq P_\theta [u]\leq q$. If (i) holds, then $P_\theta [u]=q$ has positive lower volume, and theorem 13.19.5 shows that both volumes of $u$ equal those of $q$; in particular (ii) holds, together with the final assertion.

Conversely, (ii) and $u\preceq q$ give $u\in \mathcal E_{\mathrm{vol}}(X,\theta ,q)$, and corollary 13.19.8 gives $P_\theta [u]=q$, which is (i). The same corollary shows that (i) is equivalent to (iii). By proposition 18.12.2, (iii) implies (iv), and (iv) with $F=0$ is (i). $\square$

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