I want to know if the first cohomology group of structure sheaf of grassmannian vanishes.
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3$\begingroup$ Hint: the first cohomology group of the structure sheaf of a smooth projective variety is a birational invariant. $\endgroup$– Yusuf MustopaJul 2, 2014 at 2:28
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2$\begingroup$ I suggest that you use MathStackExchange for your questions, which are not at research level. $\endgroup$– abxJul 2, 2014 at 5:25
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1 Answer
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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.
