differential with logarithmic poles

Question

where can I find the computation of the groups $H^i(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^j)$? Moreover, if $D$ is a divisor with normal crossing in $\mathbb{P}^n$, how can I compute the hypercohomology $$ \mathbb{H}^i(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^j(D)) $$ the case $D=H+I$, with $H,I$ hyperplanes could be also interesting to me (I know that the case with one hyperplane should give the cohomology of affine space). Thanks

Answer 1

The Bott formula gives the computation of $H^i(\mathbb P^n, \Omega^j (k))$. See Bott, R.: Homogenous vector bundles, Ann. of Math. 66, 203-248 (1957). In the case where you don't twist, i.e. $k=0$, it is a straightforward exercise to show $\dim ~H^i(\mathbb P^n, \Omega^j)=\delta_{ij}$, the Kronecker delta. I think it's in Hartshorne's AG, Chapter III, the section on Serre duality.

Regarding the log complex and its hypercohomology, a good place to start would be Griffiths & Harris PAG, p. 449-453. The case with two components is basically Mayer-Vietoris.