For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there is a similar formula to compute these dimensions when you consider a projective bundle instead of a projective space.



Bott formula for projective bundles
As Sasha says, the best analogy is probably the following. Let $B$ be a complex variety (or analytic space), $E$ a vector bundle on $B$ of rank $r+1$, $\pi:P=\mathbb{P}(E)\rightarrow B$ the corresponding projective bundle. Then $R^p\pi\_\*(\Omega ^q\_P)\cong \Omega \_B^{q-p}$ for $p\leq r$, $=0$ for $p>r$, and there are vanishing statements for $R^q\pi\_\*(\Omega ^q\_P(n))$. For these statements and a simple proof (if you can read french) I recommend Verdier's lecture "Théorème de Le Potier" in Astérisque no. 17 (1974).


