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I have a smooth hypersurface D in $\mathbb{P}^n$: in many books about Hodge theory (as the ones of Voisin and Carlson) they take for granted that the primitive cohomology of D is equal the variable cohomology of D. (the variable cohomology of D is the ker of the gysin map $\gamma: H^p(D,\mathbb{C}) \rightarrow H^{p+2}(\mathbb{P}^n \setminus X, \mathbb{C})$ ). it must be simple, but i can't see why

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What is $X$ here? –  Craig Westerland Oct 24 '12 at 23:04
    
i'm sorry, i meant D –  rick Oct 25 '12 at 1:25

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