Serre duality for sheaves of logarithmic differentials

Question

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?

Answer 1

To spell out Donu's comment: the wedge product $\Omega _X^p(\log D)\otimes \Omega _X^{n-p}(\log D)\rightarrow \Omega ^n_X (D)$ induces a homomorphism $w:\Omega _X^{n-p}(\log D)\rightarrow \mathcal{H}om(\Omega _X^p(\log D), \Omega ^n_X(D))$. It is an isomorphism outside $D$; to prove that it is an isomorphism it suffices to check it at a smooth point $p$ of $D$. Take coordinates $x_1,\ldots ,x_n$ so that $D$ is given by $x_1=0$ around $p$. A local basis for $\Omega _X^r(\log D)$ is given by the forms $\frac{dx_1}{x_1}\wedge dx_I$ and $dx_{J}$ with $I,J\subset [2,n]$, $\#I=r-1$ and $\#J=r$ (as usual, if $E\subset [1,n]$ consists of $e_1<\ldots <e_k$ I write $dx_E$ for $dx_{e_1}\wedge\ldots \wedge dx_{e_k}$). Now all the wedge products of an element of the first basis and an element of the second one are zero except $(\frac{dx_1}{x_1}\wedge dx_I)\wedge dx_{I^c}$ and $dx_J\wedge(\frac{dx_1}{x_1}\wedge dx_{J^c})$ which are $\pm$ the generator of $\Omega ^n_X(D)$ (here $I^c:=[2,n]\smallsetminus I$). Thus the sheaves $\Omega _X^p(\log D)$ and $\Omega _X^{n-p}(\log D)(-D)$ are Serre dual, which give your duality in cohomology.