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This question is motivated by a comment of Donu Arapura here dimension of compact support cohomology

Let $D$ be a divisor with normal crossings on some smooth projective (complex algebraic) variety $X$ of dimension $n$. Let $$ \Omega^p_X(\log D)=\Lambda^p \Omega^1_X(\log D) $$ be the sheaf of differentials forms with logarithmic poles le long de $D$.

Question: why is it true that $H^q(X, \Omega_X^p(\log D))$ and $H^{n-q}(X, \Omega_X^{n-p}(\log D)(-D))$ are Serre duals?

In general, Serre duality gives

$$ H^q(X, \mathcal{F}) \simeq H^{n-q}(X, \mathcal{F}^\vee \otimes \Omega^n_X)^\vee $$

I guess one has to use that $\Omega^n_X$ is isomorphic to $\Omega^n_X(\log D)(-D)$ but I don't see how to put things together. Can anybody help me please?

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You have posted this question here on math.SE. Please note that crossposting between multiple SE sites is highly frowned upon. Try one site first, and if you don't get a satisfactory response, ask a moderator to migrate the question to a different site. –  Zev Chonoles Jul 2 '13 at 9:33
    
Sorry, I didn't know this rule –  logoff Jul 2 '13 at 10:35
    
(Maybe I should use a bunch of one-off random user names for each comment in the future.) Anyway, your penultimate sentence is correct: you can check this by calculating in local coordinates. –  Donu Arapura Jul 2 '13 at 12:07
    
Hi, Donu. Could you please explain me how to prove your statement? I would be thankful. –  logoff Jul 2 '13 at 12:24

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