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?

migratethe question to a different site. – Zev Chonoles Jul 2 '13 at 9:33