Let $X = \mathbb{P}^{n}$ and $Z \subset X$ a smooth subvariety with $\text{dim}(Z) > 0$.
A very useful tool for calculating cohomologies in $X$ is the Bott's Formula below enunciated:
$$h^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k)) = \left\{ \begin{array}{rcl} {k+n-p \choose k}{k-1 \choose p},& \mbox{for} & q = 0,\,0\leq p\leq n, \,k> p\\ 1, & \mbox{for} & k = 0\,\ 0<p = q \leq n\\ {-k+p \choose -k}{-k-1 \choose n-p}, & \mbox{for}\,\,q = n & 0 \leq p\leq n, \,\ k < p-n \\ 0, & \mbox{otherwise} \end{array} \right. $$ In particular for $p = 0$ we have:
$$h^{q}(\mathbb{P}^{n}, \mathcal{O}_{\mathbb{P}^{n}}(k)) = \left\{ \begin{array}{rcl} {k+n\choose k} ,& \mbox{for} & q = 0, \,k \geq 0\\ {-k-1 \choose -k-1-n}, & \mbox{for} & q = n,\,\ 0 \leq p\leq n, \,\ k < p-n \\ 0, & \mbox{otherwise} \end{array} \right. $$
Let $\pi : \widetilde{X} \longrightarrow X$ be the blow up of $X$ along of $Z$.
Is Bott's formula valid in $\widetilde{X}$?
I would greatly appreciate any help.
Thank you very much.