Suppose $M$ is a complex manifold and $\Omega$ a (edit: bounded) pseudoconvex domain in $M$. Let $u:M\setminus\Omega\to\mathbb{R}$ be a pluriharmonic function. Is it true that $u$ has a pluriharmonic extension to $M$? edit: $dim\_{\mathbb{C}}(M)\geq 2$.



Extension of pluriharmonic functions
Your condition $\dim M>2$ does not save the situation: you can have many counterexamples
with $M=M'\times C^n$ where $\dim M'=1$ and your functions are independent of the second
variable. And in dimension $1$ you certainly have plenty of pluriharmonic (=harmonic) functions which do not extend anywhere.


