# An analogue of Lefschetz hyperplane theorem for complements to subvarieties in $\mathbb C^n$ ?

Let $V^{2k}$ be a complex subvariety of dimension $2k$ (real dimension $4k$) in $\mathbb C^n$. Let $A$ be a complex $n-k$ dimensional plane in $\mathbb C^n$.

Question. Is it true that the inclusion $H_{2n-2k-1}(A\setminus (V\cap A))\to H_{2n-2k-1}(\mathbb C^n\setminus V)$ is injective?

We don't require $V^{2k}$ to be smooth, but $V^{2k}$ must be equidimesional, i.e. all its irreducible components have dimension $2k$.

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I was about to direct you to Katz's article in Motives I but... oops, you are not asking for affine Lefschetz hyperplane! – shenghao May 10 '11 at 23:28

Yes. Indeed all irreducible components of $V\cap A$ have positive dimension. So the map is injective, since $H_{2n-2k-1}(A\setminus (V\cap A))=0$ as is shown in the answer to the following question:
A bound on the top homology of a complement to a variety in $\mathbb C^n$