# deducing Bott vanishing

In the book of Okonek et al. on vector bundles it is suggested as an exercise to derive the dimensions of cohomology $H^q(\mathbb{P}^n, \Omega^p)$, using Euler sequence and Serre duality, from the vanishing of $H^q(\mathbb{P^n}, \mathcal{O}_{\mathbb{P}^n})$ when $q > 0$. The latter is claimed to hold, with a reference to the book of Banica and Stanasila, through "a clever Laurent separation". Now, while this cohomology vanishing can be deduced over the complex numbers via Serre duality and computation of Hodge numbers for $\mathbb{P}^n$, I am curios as to what could have been meant by this "Laurent separation" trick, and if the argument implied is independent of Hodge theory. As there is no precise reference in the Banica and Stanasila's book, I am at a loss.

Any guesses at what this remark could mean?

• I think they are referring to Chapter IV, Lemma 1.1, on page 139 of the English edition. The argument there is attributed to Frenkel, and this is probably referring to Proposition 33.1 in Cohomologie non abélienne et espaces fibrés. Dec 8, 2017 at 4:41
• @TakumiMurayama: thanks! if you just copy-and-paste your comment as an answer I'd gladly accept it and close the question Dec 8, 2017 at 13:53