I'm working on some problem in algebraic geometry. I need a reference to the following result:

Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$ be a non zero polynomial. The complement manifold $\mathbb{C}^{h}\setminus\left\lbrace F=0\right\rbrace$ is a nonempty open connected subspace of $\mathbb{C}^{h}.$

Probably this is contained in some old work of Zariski (or even older). Please do not esitate to suggest me some bibliographical references.


2 Answers 2


Claim: The complement $U=\mathbb C^h\setminus \{F=0\}$ is path-connected and thus connected.
Given $a,b\in U$ consider the affine complex line $L_{a,b}=L$ joining $a$ to $b$.
The polynomial $F\mid L$ is not zero since it is not zero at $a$ nor at $b$.
Thus it has only finitely many zeros on $L$ ($\cong \mathbb C $ !) and we can find a path from $a$ to $b$ in $L$ avoiding these zeros: that path is contained in $U$.


$F$ is non zero, thus the complement is not empty. The regular part of the zero set, namely $\{z:F(z)=0, dF(z)\ne 0\}$, has complex codimension 1, thus real codimension 2; so the complement of this is connected. The singular part $\{z:F(z)=0, dF(z)=0\}$ is of higher codimension. There are only finitely many such parts, thus the complement is connected.


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