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Let $D$ be a simple normal crossing divisor on a smooth projective variety over a field $k \subset \mathcal{C}.$ Write $D_i$ with $i \in I$ for its irreducible components. Denote, as usual,

$D_J=\cap_{i \in J} D_i$

for any subset $J \subset I$. By assumption, all $D_J$ are smooth. Is is true that $H^*(D)$ (Betti or de Rham cohomology) is generated by $H^\ast(D_J)$ for all $J \subset I$?

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This is not true, consider for example a divisor $D$ on a surface that is a wheel of $\mathbb P^1$'s, i.e, each $\mathbb P^1$ intersects two neighbouring $\mathbb P^1$'s. Then $\pi_1(D)=\mathbb Z$, so $H^1(D)=\mathbb Z$. –  Dmitri Mar 23 '13 at 13:56
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