Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $2k$.

share|improve this question
    
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

1 Answer 1

up vote 1 down vote accepted

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 $\mathbb C^n$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.