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Say I have two locally free sheaves $F,G$ on projective variety $X$. I know the cohomology groups $H^i(X,F)$ and $H^i(X,G)$. Is this enough to give me information about $H^i(X,F\otimes G)$? In particular, if $H^i(X,F)=0$, what conditions on $G$ guarantee that also $H^i(X,F\otimes G)=0$?

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    Have you seen http://mathoverflow.net/questions/34673/kunneth-formula-for-sheaf-cohomology-of-varieties?2010-11-10
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    Just for reference you'll need more than both vanishing to get that the tensor product vanishes. Take for instance $X=\mathbb{P}_k^n$, $F=\mathcal{O}(-n)$, and $G=\mathcal{O}(-1)$, then $H^n(X, F)=0$, $H^n(X, G)=0$ but $H^n(X, F\otimes G)=k$.2010-11-10

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