{title2=$H^q(\mathbb P^n,\mathcal O(d))$}

For every integer $d$, the intermediate cohomology of a twisting sheaf vanishes:
$$
H^q(\mathbb P_k^n,\mathcal O(d))=0\qquad(0<q<n).
$$
The zeroth cohomology for $d\geq0$ consists of degree-$d$ homogeneous polynomials, and the top cohomology is determined by <Serre duality>.


Cohomology of twisting sheaves on projective space

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{wiki}

For a smooth projective $n$-dimensional variety $X$ over a field and a coherent sheaf $\mathcal F$, Serre duality gives a perfect pairing between $H^q(X,\mathcal F)$ and $\operatorname{Ext}^{n-q}(\mathcal F,\omega_X)$, where $\omega_X$ is the canonical sheaf.


Codex @codex  0

 Incoming links: Serre duality