ag.algebraic geometry - Genus of Grassmannians and Flag Manifolds

EDIT: My first answer was confusing and not quite accurate. Let me try again.

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.

Now using Serre duality, we conclude that only the top cohomology of the canonical sheaf is nonzero, which means that the geometric genus is also 0.