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Let $X$ be a complex smooth projective variety and let $\iota: Y \hookrightarrow X$ be a smooth hyperplane section. Put $\dim(X)=n+1$. Then weak Lefschetz says that $$ \iota^\ast: H^k(X^{an}, \mathbb{Q}) \to H^k(Y^{an}, \mathbb{Q}) $$ is an isomorphism for $k \leq n-2$.

I would be interesting in the following variant:

  • First, instead of considering the whole $X$, I want to look at $U=X-D$, where $D$ is a simple normal crossings divisor.

  • Secondly, instead of taking $\mathbb{Q}$ as coefficients, I would like to look at a rank one local system $V$ of $\mathbb{C}$-vector spaces on $U^{an}$.

Assume $Y$ is a smooth section of $X$ (intersecting properly all the intersections of the irreducible components of $D$). Put $W=Y-D \cap Y$. Is it true that

$H^k(U^{an}, V) \to H^k(W^{an}, V_{|W^{an}})$

is an isomorphism for $k \leq n-2$? Same question for cohomology with compact support.

share|improve this question
    
Lefschetz theorem also says that the inclusion map $i: Y\to X$ is $n$-connected (if $X$ is $n+1$-dimensional), see e.g. Milnor's book on Morse Theory. Thus, $i$ induces an isomorphism of cohomology with with coefficients in arbitrary flat bundle/locally constant sheaf (up to degree $n-1$), since the latter can be computed by looking at the skeleta of dimension $\le n$ of cell complexes for $X$ and $Y$. –  Misha Apr 10 '13 at 23:35
    
Thanks for your answer Misha! Do you have a reference where your statement about cohomology is proved? Remark that my connection has logarithmic singularities on $D$. Is everything still working? –  lefsloc Apr 11 '13 at 8:37
    
If you want log singularities, you could perhaps proceed like this: first look up the results of Esnault-Viehweg on generalizations with log. sing. of the Kodaira vanishing theorem and of the degeneration of the Hodge-de-Rham spectral sequence (maybe in their book ?) and then replicate the proof of the weak Lefschetz theorem given in the book of Griffiths and Harris (p. 156, chap. I, §2). –  Damian Rössler Apr 11 '13 at 11:09
    
Thanks Damian. That sounds great –  lefsloc Apr 11 '13 at 13:27
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