Theorem 9.10.13 (Fornaess–Narasimhan)48473380B913

Theorem 9.10.13 (Fornaess–Narasimhan).48473380B913 Let $\varphi \colon X\rightarrow [-\infty ,\infty )$ be a function. Assume that $\varphi $ is not identically $-\infty $ on any irreducible component of $X$. Then the following are equivalent:

1. C2704BBBABCF

   $\varphi $ is psh;

2. 917F8CC8F375

   $\varphi $ is usc and for any morphism $f\colon \Delta \rightarrow X$ from the open unit disk $\Delta $ in $\mathbb {C}$ to $X$ such that $f^*\varphi $ is not identically $-\infty $, the pull-back $f^*\varphi $ is subharmonic.