# Genus of Grassmannians and Flag Manifolds

In light of the answers given to this question, I would like to pose a more general one: Do all Grassmannian spaces have genus 0? If so, do there exist any flag manifolds with non-zero genus?

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In arbitrary characteristic, the structure sheaf of any homogeneous space $G/P$ (for $G$ a semisimple group) has no higher cohomology. This is an instance of Kempf's vanishing theorem. The space of sections is 1-dimensional, so this implies that the arithmetic genus is 0.
Steven, do you mean "$\mathit{complete}$ homogeneous space", i.e. $G/P$? That's certainly implied by the question, but the answer is a but unclear. –  Victor Protsak Jun 12 '10 at 4:08