# Bott's Formula for Grassmannians

Bott's Formula gives the dimension of the cohomology $H^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k))$ of the $k$-twisted sheaf of $p$-differential forms on the projective space $\mathbb{P}_{\mathbb{C}}^{n}$. I was wondering if there is a similiar formula for complex Grassmannians $\mathbb{G}(r, n)$ instead of projective spaces $\mathbb{P}_{\mathbb{C}}^{n}$. I searched for a while and I only found results that provide necessary and sufficient conditions for the vanishing of such cohomology groups. When such cohomology groups are non-zero I did not find any results providing the dimension of such cohomologies.

Do you know if there is any formula for the dimension of the cohomology groups $H^{q}(\mathbb{G}(r, n), \Omega_{\mathbb{G}}(k))$ as $\mathbb{C}$-vector spaces? Where could I find it?

Thank you very much!

• I don't know what a $k$-twisted sheaf is, but perhaps you are looking for something like this Bott-Borel-Weil formula? en.wikipedia.org/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem Grassmanian is of the form $G/P$ for a certain subgroup $P$ containing $B$ (these are called parabolic subgroups). Generalization of the BBW formula to this case is in an old famous paper by Kostant. All of this is neatly packaged in a monograph Parabolic geometries by Čap and Slovák. Aug 18, 2014 at 12:45

For $r = 1$ in Lemma 0.1 of this paper