Lemma 13.28.12422DD1AB3AB

Proof. The bounds for the Green kernel give $w_\mu \in L^1(Y)$ and

In particular $w_\mu $ is $\theta $-plurisubharmonic, and its polar curvature is $\mu $. We claim that, for every $v\in \mathrm{PSH}(Y,\theta )$,

The forward implication follows from corollary 5.27.15, which gives $T_ v\geq \mu _ v\geq \mu _{w_\mu }=\mu $. Conversely, if $T_ v\geq \mu $, the distribution $v-w_\mu $ satisfies $\mathrm{dd}^{\mathrm{c}}(v-w_\mu )\geq -(A-\mu (Y))\eta $, so its quasi-subharmonic representative is bounded above on the compact curve $Y$. By lemma 5.24.3, $v\preceq w_\mu $.

Thus $p_\mu $ is the upper envelope of all $v\leq 0$ with $T_ v\geq \mu $. This family is closed under maxima by (5.27.16). By Choquet's lemma the envelope is the limit of an increasing sequence of such maxima, which converges in $L^1$; hence the current inequality $T_{p_\mu }\geq \mu $ passes to the limit. Also $p_\mu \geq w_\mu -\sup _ Yw_\mu $. Hence $b=p_\mu -w_\mu $ has a quasi-subharmonic representative that is bounded below, and compactness bounds it above. Therefore $p_\mu \sim w_\mu $. The positive measure

charges no polar set because $b$ is bounded. This proves $\mu _{p_\mu }=\mu $ and $U(p_\mu )=A-\mu (Y)$, and ordinary equivalence gives $P_\theta [p_\mu ]=P_\theta [w_\mu ]=p_\mu $.

For arbitrary $u$, take $\mu =\mu _ u$. Then $T_ u\geq \mu $, so (13.28.3) gives $u\preceq p_\mu $, and $U(u)=A-\mu (Y)=U(p_\mu )$. If this mass is positive, corollary 13.19.8 gives (13.28.2). If it is zero, then $\mu (Y)=A$ and $T_ u=\mu =T_{w_\mu }$. The difference $u-w_\mu $ is then harmonic on $Y$ and hence constant, which gives the same conclusion. Finally, every ordinary model $p$ equals $P_\theta [p]=p_{\mu _ p}$, and distinct polar measures give distinct models because $\mu _{p_\mu }=\mu $. $\square$