Example 17.10.62F494BEED8F5

Example 17.10.6.2F494BEED8F5 Let $\varphi _0\in \mathrm{PSH}(X,\theta )$, $C\in \mathbb {R}$. Let

Then $(\varphi _ t)_{t\in [0,1]}$ is a subgeodesic. Indeed, since $\operatorname{Re}z$ is harmonic in $z$, the complexification $\Phi (x,z)=\varphi _0(x)+C\, \operatorname{Re}z$ is $p_1^*\theta $-psh.

As a consequence, the constant $(\varphi _0)_{t\in [0,1]}$ is a subgeodesic, called the constant subgeodesic at $\varphi _0$.

A more general version is as follows: Suppose that $(\varphi _ t)_{t\in [0,1]}$ is a subgeodesic in $\mathrm{PSH}(X,\theta )$, $C_1,C_2\in \mathbb {R}$. Then $(\varphi _ t+C_1t+C_2)_{t\in [0,1]}$ is also a subgeodesic.