PomeEECB2B1DF1A4Corollary 19.9.17.EECB2B1DF1A4 For every holomorphic line bundle $L$ on a connected compact complex manifold $X$ of dimension $n$,
If $X$ is not Moishezon, then, for each $L$, either all positive-degree section spaces of $L$ vanish, or there is an integer $d<n$ for which
In particular $m^{-n}h^0(X,L^ m)\to 0$. These conclusions apply when $X$ fails the bounded mass property.
Proof. If all positive-degree section spaces vanish, there is nothing to prove. Otherwise, apply the theorem to the full section ring. Since $W_ m=0$ for $e\nmid m$, the limit gives $h^0(X,L^ m)=O(m^ d)$ for all degrees, with $d\leq n$. If $d=n$, the section-ratio field contains $n$ algebraically independent meromorphic functions on $X$, so $X$ is Moishezon. Thus $d<n$ when $X$ is not Moishezon. Finally, by Moishezon's theorem a Moishezon manifold admits a modification from a projective, hence Kähler, manifold, so it has the bounded mass property by lemma 19.9.12. The contrapositive gives the last assertion. $\square$
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