# Cohomology of Line Bundles

by Tarik Rian
PhD Dissertations | September 2007
Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p$ and $B$ a Borel subgroup of $G$. Each character $\lambda$ of $B$ induces a line bundle $\mathcal{L}(\lambda)$ on the flag variety $G/B$, and the cohomology group $H^\bullet (G/B,\mathcal{L}(\lambda))$ has a natural $G$-structure. The vanishing behaviour of all such cohomology depend on the characteristic of $k$. When $\operatorname{char}(k)=0$, the Borel-Borel-Weil theorem gives a complete description of the vanishing behaviour whereas the corresponding problem in prime characteristic is wide open. In this thesis we collect some well-known results about the vanishing behaviour of these cohomology groups, and at the same time we will demonstrate that many of these results have analogues in the corresponding quantum case. Related to the problem of describing the cohomology group of line bundles on $G/B$ is the calculation of the $B$-cohomology of $1$-dimensional $B$-modules. When $\operatorname{char}(k) =0$, there is an easy well-known description of this cohomology whereas the corresponding problem in prime characteristic is wide open. We develop some new techniques which enable us to calculate all such cohomology in degrees at most $3$ when $p$ is larger than the Coxeter number for $G$. Our methods also apply to the corresponding question for quantum groups at roots of unity. Furthermore, when $\operatorname{char}(k) =0$, we will calculate the fourth cohomology group explicitly. But this requires a different argument than the one given in the modular case.
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