TITLE: Sheaf cohomology of projective spaces
QUESTION [2 upvotes]: I came across Bott's formula in "Vector bundles on complex projective spaces" by Okonek, Schneider & Spindler.
The formula is a formula for $h^q(\mathbb P^n,\Omega^p(k))$, where $\Omega^p(k)$ is the $k$-twisted sheaf of sections in the p-th power of the cotangent bundle of $\mathbb P^n$.
Is there a duality I can use for deriving the value of, say $h^0(\mathbb P^3,\bigwedge^2 T_{\mathbb P^3})$?

REPLY [2 votes]: For a 3 dimensional vector space $V$ one has a canonical nondegenerate pairing $V \otimes \bigwedge^2 {V} \rightarrow detV$. Using the general principle that everything that works canonically for vector spaces works for vector bundles, take $V = T_{\mathbb P^3}$. We have $detV=\omega_{\mathbb P^3}^*$, where $\omega_{\mathbb P^3}$ is the canonical sheaf. It is standard that $\omega_{\mathbb P^n} \cong O(-n-1)$ (see Hartshorne II.8, for example), so we have 
$\bigwedge^2 {V} \cong Hom_{O_{\mathbb P^3}}(V,O(4))\cong Hom_{O_{\mathbb P^3}}(V(-4),O_{\mathbb P^3}) \cong V^*(4)$.
Thus, $h^0(\mathbb P^3,\bigwedge^2 T_{\mathbb P^3}) = h^0(\mathbb P^3, \Omega_{\mathbb P^3}(4))$, for which you say you have a formula.
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