Lemma 17.8.23D0F1309E327

Proof. Let $Z$ be either $X\times S$, on which the group $\Gamma =\mathbb R$ acts by the imaginary translations $(x,\zeta )\mapsto (x,\zeta +\mathrm iy)$, or $X\times A$ with $A=\{ \mathrm e^{-1}<|z|<1\} $, on which $\Gamma =\mathbb R$ acts by the rotations $(x,z)\mapsto (x,\mathrm e^{\mathrm iy}z)$, according to where $D$ lies. Put $\tau (x,\zeta )=\operatorname{Re}\zeta $ in the first case and $\tau (x,z)=-\log |z|$ in the second, and let $\theta _ Z$ be the pullback of $\theta $ to $Z$. The action of $\Gamma $ preserves $\theta _ Z$ and $\tau $, and the $\Gamma $-invariant functions on $Z$ are exactly the functions whose value at $(x,w)\in Z$ is $v_{\tau (x,w)}(x)$, for a family $(v_ t)_{0<t<1}$ of functions on $X$; by the discussion following definition 17.7.1, such a function is $\theta _ Z$-psh if and only if the complexification of $(v_ t)$ on $X\times S$ is $p^*\theta $-psh. Let $U$ be the $\Gamma $-invariant function on $Z$ defined by the weak geodesic $(G_ t)$; it is $\theta _ Z$-psh and bounded above by $\max \{ \sup _ Xu_0,\sup _ Xu_1\} $, since $G_ t\leq (1-t)u_0+tu_1$.

Now $D\subset Z$ and $\mathrm{dd}^{\mathrm{c}}a=\theta _ Z$ on $D$, so that $h=U+a$ is plurisubharmonic on $D$; we show that it is maximal. Let $v$ be plurisubharmonic on $D$ with $v\leq h$ outside a compact set $K\subset D$. Every boundary point of $D$ in $Z$ has a neighborhood disjoint from $K$, on whose intersection with $D$ we have $v-a\leq U$; since $U$ is upper semicontinuous, the boundary condition of proposition 5.9.5 holds, and the function $\widetilde U$ equal to $\max \{ U,v-a\} $ on $D$ and to $U$ elsewhere is $\theta _ Z$-psh (apply the proposition in charts after adding smooth local potentials of $\theta _ Z$). Moreover $\widetilde U\geq U$, and $\widetilde U=U$ outside $K$. Since $v-a$ is bounded above on the compact set $K$, the function $\widetilde U$ is bounded above.

The functions $\widetilde U\circ \gamma $, $\gamma \in \Gamma $, are $\theta _ Z$-psh and uniformly bounded above, so their regularized upper envelope $\breve U$ is $\theta _ Z$-psh (theorem 5.10.2, applied in charts after adding smooth local potentials); it is $\Gamma $-invariant and satisfies $\breve U\geq \widetilde U$. The compact set $\tau (K)$ lies in an interval $[r_1,r_2]$ with $0<r_1\leq r_2<1$, so the $\Gamma $-orbit of $K$ lies in $\{ r_1\leq \tau \leq r_2\} $. Outside this set every $\widetilde U\circ \gamma $ equals $U\circ \gamma =U$, so $\breve U=U$ on the open set $\{ \tau <r_1\} \cup \{ \tau >r_2\} $, because $U$ is upper semicontinuous. Therefore $\breve U$ is defined by a subgeodesic whose slices coincide with $G_ t$ for $t<r_1$ and for $t>r_2$; in particular its endpoint potentials are $u_0$ and $u_1$. It is a competitor for the Perron segment between $u_0$ and $u_1$, so $\breve U\leq U$. Consequently $\widetilde U\leq U$, so $v-a\leq U$ on $D$, that is, $v\leq h$. This is maximality in the sense of definition 7.7.6. $\square$