Let $V$ be a subvariety of $\mathbb C^n$ with irreducible components of dimension >$0$. Is $H\_{2n-1}(\mathbb C^n\setminus V)=0$? 



A bound on the top homology of a complement to a variety in $\mathbb C^n$
It's certainly true, though I don't know the best way of proving it.


One way to see it is that if you take the one-point compactifications $V^+\subset(\mathbb{C}^n)^+=S^{2n}$, you get a connected space, since all components contain the point at infinity. The result then follows by Alexander duality in $S^{2n}$.


