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Let $X$ be a compact complex n-fold . Then for every coherent sheaf $\mathfrak{F}$ on $X$ , and every holomorphic line bundle $L$ on $X$ , then the dimension of $H^0 (X,\mathfrak{F}\otimes\mathcal{O}_X(L))$ does not depend on $L$ when dim Supp$\mathfrak{F}=0$ .

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Context? Motivation? Some comments to illustrate which cases you can already do, what you know, etc? –  Yemon Choi May 8 '11 at 1:47

1 Answer 1

up vote 10 down vote accepted

$\dim\mathrm{Supp}\\, \mathfrak F=0$ implies that $\mathfrak F\otimes \mathscr O_X(L)\simeq \mathfrak F$ and hence its cohomology is independent of $L$.

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how to get this isomorphism ? –  HKSHLZW May 8 '11 at 9:06
The support of $\mathfrak{F}$ is just a finite set of closed points. Now choose trivializations of $L$ around each point to get the desired isomorphism. –  Martin Brandenburg May 8 '11 at 9:26
Thank you very much ! –  HKSHLZW May 8 '11 at 9:31

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