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Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample.

I am trying to follow the proof of Theorem 2.1 of http://arxiv.org/pdf/math/0404341v1.pdf, where the author claims the following.

"surjectivity of $π_i(D_i − \cup_{j\neq i} D_j) \to π_i(X − D)$ ... follows from ampleness of the components $D_i$."

Based upon the statement of theorem 2.1, I presume that the author means in the range $2 \leq i \leq n-1$.

There are a few things I don't understand about this statement.

1) The first is what is the map on $\pi_i$ that the author is proposing?

2) Second, what version of the Lefschetz hyperplane is the author invoking ? I know that for experts there are many Lefschetz hyperplane theorems (e.g. affine Lefschetz hyperplane theorem due to Hamm, etc.). Perhaps if one has a deep enough understanding of the theorem this result is trivial. For me, who just understands the classical version in say Milnor's book, this is not so clear. Can anyone please explain a bit more of what the author is asserting and from what results in the literature it follows?

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1 Answer 1

I think this follows from the following result, that can be found in Dimca's book Singularities and Topology of Hypersurfaces, page 25.

Zariski Theorem of Lefchetz Type. Let $X \subset \mathbb{P}^n$ be a closed subvariety which is locally a complete intersection of dimension $m$. Let $Z$ be another closed subvariety of $\mathbb{P}^n$ and let $\mathscr{A}$ be a Withney stratification of $X$ such that $X \cap Z$ is a union of strata of $\mathscr{A}$. Then the inclusion $$j_H \colon (X \setminus Z)\cap H \longrightarrow X \setminus Z$$ is a $(m-1)$-equivalence for any hyperplane $H \subset \mathbb{P}^n$ which is transversal to all the strata in $X \cap Z$. In other words the induced map in homotopy $$\pi_i (j_H) \colon \pi_i ((X \setminus Z)\cap H) \longrightarrow \pi_i(X \setminus Z)$$ is an isomorphism for $i < m$ and an epimorphism for $i=m$.

Now just apply this by taking $$Z= D \quad \textrm{and} \quad H=D_i.$$ This works since $D_i$ ample means that $aD_i$ becomes a hyperplane section of $X$ by taking the corresponding Veronese embedding (for $a \in \mathbb{N}$ large enough).

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Hi Francesco, thank you very much for your suggestion. I am a bit worried about the following point $H=D_i$ doesn't seem to be transverse to X \cap Z. Moreover, I don't think the conclusion of the theorem you have stated can hold for $\pi_1$ if we consider e.g. $P^2$ and two generic hyperplanes $D_1$ and $D_2$? Though maybe I misunderstand the idea. –  user36931 Jul 22 '13 at 13:45
    
I should be clear and say that I don't think it holds if $Z=D_1\cup D_2$ and $H=D_1$ –  user36931 Jul 22 '13 at 13:54
    
uhm...right. However, I still think that this is the result that one needs, even if it does not apply as directly as I (hurriedly) wrote. –  Francesco Polizzi Jul 22 '13 at 14:23

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