Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.