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

Question

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 $2k$.

Answer 1

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 $\mathbb C^n$