2
$\begingroup$

Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful compactification)?

In characteristic $0$, Syu Kato proves this result in this paper, working over $\mathbb{C}$.

In characteristic $p$, it is known that higher cohomologies vanish for line bundles corresponding to dominant weights, for example as in Theorem 3.2 in this paper.

Thank you very much!

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ The "usual" Borel-Weil-Bott theorem on varieties $G/B$ fails in positive characteristic, and $G/B\times G/B$ sits inside the wonderful compactification $\widehat{G}$ as the common intersection of all boundary divisors. So my intuition suggests that Borel-Weil-Bott also fails for $\widehat{G}$ in positive characteristic. $\endgroup$
    – Jason Starr
    Commented Jul 7, 2022 at 11:01

0

Reset to default

You must log in to answer this question.