I want to know if the first cohomology group of structure sheaf of grassmannian vanishes.



what is the first cohomology group of structure sheaf of grassmannian
Over the complex numbers, the Grassmanian $X = G(n,k)$ is simply-connected. Since this also a smooth projective variety, it is compact and Kaehler, so we have $0 =b\_1(X) = 2h^{1,0}(X)$, where $b\_1(X)$ denotes the first Betti number and $h^{1,0}(X) = \text{dim}\ H^1(\mathcal{O}\_X)$. This gives the vanishing that you seek. 


