Definition 17.10.28699FCF63094

Definition 17.10.2.8699FCF63094 Let us fix $\varphi _0,\varphi _1\in \mathrm{PSH}(X,\theta )$. A subgeodesic from $\varphi _0$ to $\varphi _1$ is a family $(\varphi _ t)_{t\in (0,1)}$ in $\mathrm{PSH}(X,\theta )$ such that

1. 9DAD7522B37E

   if we define

   \[ \Phi \colon X\times \Bigl \{ z\in \mathbb {C}:\operatorname{Re}z\in (0,1) \Bigr \} \rightarrow [-\infty ,\infty ),\quad (x,z)\mapsto \varphi _{\operatorname{Re}z}(x), \]

   then $\Phi $ is $p_1^*\theta $-psh, where $p_1\colon X\times \{ z\in \mathbb {C}:\operatorname{Re}z\in (0,1)\} \rightarrow X$ is the natural projection;

2. 80E0ABF2B86F

   when $t\to 0+$ (resp. $t\to 1-$), $\varphi _ t$ converges to $\varphi _0$ (resp. $\varphi _1$) with respect to the $L^1$-topology.

We also say $(\varphi _ t)_{t\in [0,1]}$ is a subgeodesic.

We call $\Phi $ the complexification of the subgeodesic $(\varphi _ t)_ t$.