Definition 17.10.17872270015CE7

Definition 17.10.17.872270015CE7 A ray $\ell =(\ell _ t)_{t\geq 0}$ is a subgeodesic ray in $\mathrm{PSH}(X,\theta )$ if for any $0\leq a\leq b$, the segment $(\ell _{t})_{t\in [a,b]}$ is a subgeodesic in $\mathrm{PSH}(X,\theta )$. We say $\ell $ emanates from $\ell _0$.

The complexification of a subgeodesic ray $\ell $ is defined as the potential

Note that $\Phi $ is $p_1^*\theta $-psh, where $p_1\colon X\times \{ z\in \mathbb {C}:\operatorname{Re}z>0\} \rightarrow X$ is the natural projection.