Lemma 17.7.5F4AAE999A6C4

Proof. Let $0\leq a<b\leq 1$. The restriction of $(u_ t)$ to $[a,b]$, reparametrized as $t\mapsto u_{a+(b-a)t}$, has the complexification $U\circ F$, where $F(x,\zeta )=(x,a+(b-a)\zeta )$ is a holomorphic map of $X\times S$ into itself with $p\circ F=p$. In a chart $D$ of $X$ on which $\theta =\mathrm{dd}^{\mathrm{c}}h$, we have $(U+h\circ p)\circ F=U\circ F+h\circ p$, which is plurisubharmonic on $D\times S$ by proposition 5.9.3; it is not identically $-\infty $, because its slices are the functions $u_{a+(b-a)t}+h$, which are not identically $-\infty $ by lemma 17.7.3. Hence $U\circ F$ is $p^*\theta $-psh, and the restriction is a subgeodesic with endpoint potentials $u_ a$ and $u_ b$, by the $L^1$ continuity of proposition 17.7.4. An affine function $c_0+c_1\operatorname{Re}\zeta =\operatorname{Re}(c_0+c_1\zeta )$ is pluriharmonic, so adding it preserves $p^*\theta $-plurisubharmonicity, and the change of background replaces $U$ by $U-g\circ p$ and $p^*\theta $ by $p^*\theta +\mathrm{dd}^{\mathrm{c}}(g\circ p)$; this proves the first assertion.

For the replacement, let $U$ be the complexification of the given subgeodesic, and $V$ that of the new one on $W=X\times \{ a<\operatorname{Re}\zeta <b\} $. We verify the boundary condition of proposition 5.9.5 along each seam $\operatorname{Re}\zeta =a$ or $\operatorname{Re}\zeta =b$ lying in $S$; consider $a>0$. Let $(x_ k,\zeta _ k)\in W$ converge to $(x,\zeta )$ with $\operatorname{Re}\zeta =a$, and put $t_ k=\operatorname{Re}\zeta _ k$. Denote the slices of the new subgeodesic by $v_ t$, $a<t<b$; they converge to $u_ a$ in $L^1(X)$ as $t\downarrow a$.

We show that $\varlimsup _ kV(x_ k,\zeta _ k)\leq U(x,\zeta )=u_ a(x)$. Since $v_{t_ k}\to u_ a$ in $L^1(X)$, lemma 17.4.3 shows that the tail envelopes $w_ l=(\sup _{k\geq l}v_{t_ k})^*$ are $\theta $-psh and decrease pointwise to $u_ a$. For $k\geq l$ we have $V(x_ k,\zeta _ k)=v_{t_ k}(x_ k)\leq w_ l(x_ k)$, and $w_ l$ is upper semicontinuous, so $\varlimsup _ kV(x_ k,\zeta _ k)\leq w_ l(x)$ for every $l$. Letting $l\to \infty $ proves the claim. A seam $\operatorname{Re}\zeta =b$ with $b<1$ is treated in the same way, with $t_ k\uparrow b$. Hence, by proposition 5.9.5 applied in charts after adding smooth local potentials of $p^*\theta $, the function equal to $\max \{ U,V\} =V$ on $W$ and to $U$ elsewhere is $p^*\theta $-psh. Its endpoint limits at $t=0$ and $t=1$ are those of the given subgeodesic, or those of the new one if $a=0$ or $b=1$; in either case they are $u_0$ and $u_1$. $\square$