# what is the first cohomology group of structure sheaf of grassmannian [closed]

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

• Hint: the first cohomology group of the structure sheaf of a smooth projective variety is a birational invariant. Jul 2, 2014 at 2:28
• I suggest that you use MathStackExchange for your questions, which are not at research level.
– abx
Jul 2, 2014 at 5:25

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.