Let $X$ be a smooth compact complex manifold of dimension $n$. Suppose $L$ is a line bundle on $X$ such that $dim(H^0(X,L^k))>c\cdot k^n$ for $c>0$ and $k>>0$.
Question. Is it true that $X$ is Moishezon? Is there some reference for this statement?
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$\begingroup$ do you mean $k^n$? $\endgroup$– Chen JiangCommented Jul 19, 2017 at 15:05
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9$\begingroup$ The question correspond to volume of line bundle. $L$ is big, implies the projectivity of X is Moishezon. Note that $L$ is big iff $Vol(L)>0$. By definition $$Vol(L)=\lim sup_{p\to\infty}\frac{n!}{p^n}dim H^0(X,L^p)$$ $\endgroup$– user21574Commented Jul 19, 2017 at 15:30
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3$\begingroup$ You may see 2.2.15 of the following charpter: webusers.imj-prg.fr/~xiaonan.ma/M2_17/Moishezon.pdf $\endgroup$– Chen JiangCommented Jul 19, 2017 at 15:31
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2$\begingroup$ @ChenJiang: why not turn your good comment into an answer? That will be helpful for future visitors, prevent against link-rot, etc. $\endgroup$– Lazzaro CampeottiCommented Jul 21, 2017 at 10:10
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