Let $U$ be

An answer to the open set of ${\bf C}^n$ defined by $P\neq 0$. Take two points $a$ and $b\in U$more general question could be in three steps.To show that

1) If $U$ X$ is a complex algebraic variety, connected for the Zariski topology, then it suffices to show that the intersection $L\cap U$ is connected , where $L$ is the line $(ab)$. Write a point $p$ of for the line as $a+tu$, where $u=b-a$ usual topology, an important and $t\in{\bf C}$. This describes $L\cap U$ as the complement nontrivial fact that can be found in ${\bf C}$ of the locus defined by the equation $P(a+tu)=0$; the latter is finiteMumford's Red Book, or in Shafarevich's one.

So it remains to show that the

2) The complement to a finite set of point closed Zariski subset in the plane an irreducible variety is connected . This is almost obvious: try to go straight away from $a$ to $b$ and when you hit one of for the removed points, make a small detour (a half circleZariski topology.

3) The varieties $SL(n,{\bf C})$ and resume to your straight path.$SL(N,{\bf C})^N$ are irreducible

Let $U$ be the open set of ${\bf C}^n$ defined by $P\neq 0$. Take two points $a$ and $b\in U$. To show that $U$ is connected, it suffices to show that the intersection $L\cap U$ is connected, where $L$ is the line $(ab)$. Write a point $p$ of the line as $a+tu$, where $u=b-a$ and $t\in{\bf C}$. This describes $L\cap U$ as the complement in ${\bf C}$ of the locus defined by the equation $P(a+tu)=0$; the latter is finite.

So it remains to show that the complement to a finite set of point in the plane is connected. This is almost obvious: try to go straight away from $a$ to $b$ and when you hit one of the removed points, make a small detour (a half circle) and resume to your straight path.