If $X$ is an irreducible variety in $\mathbb{C}^n$, then $\mathbb{C}^n \setminus X$ is path-connected.

Question

My question: Prove that if $X$ is an irreducible variety in $\mathbb{C}^n$, then $\mathbb{C}^n \setminus X$ is path-connected in the standard metric topology.

For every two distinct points $p,q$ in $\mathbb{C}^n \setminus X$,let $L$ denote the line through $p$ and $q$. I can prove the line is a continuous path but I don't know how to show $L \subseteq \mathbb{C}^n \setminus X$ i.e. how to use the condition that $X$ is irreducible?

I'm stuck on this question for several hours. Can anyone give me a hint? Thank you !

Answer 1

Hint: Think about the complex line $\{tp+(1-t)q\in\mathbb{C}^n\mid t\in\mathbb{C}\}\cong\mathbb{C}$.