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Let $D$ be a normal crossings divisor on some smooth projective variety $X$ (say over the complex numbers) and let $\Omega^p_X(\log D)$ be the sheaf of logarithmic $p$-forms. It is $$ \Lambda^p \Omega^1_X(\log D) $$

Can somebody explain why Serre duality gives $$ H^q(X, \Omega^p_X(\log D)) \simeq H^{n-q}(X, \Omega^{n-p}_X(\log D)(-D))? $$

Thanks for your help

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Crossposted on MathOverflow. –  Zev Chonoles Jul 2 '13 at 9:36

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