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

  • 3
    $\begingroup$ Hint: the first cohomology group of the structure sheaf of a smooth projective variety is a birational invariant. $\endgroup$ Jul 2, 2014 at 2:28
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    $\begingroup$ I suggest that you use MathStackExchange for your questions, which are not at research level. $\endgroup$
    – abx
    Jul 2, 2014 at 5:25

1 Answer 1


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.


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