Pome14480FFA5B57Corollary 19.9.20.14480FFA5B57 If $S_ h$ is empty, then $H^0(X,L^ p)=0$ for every $p\geq 1$. Otherwise
In this second case, every graded section algebra with a nonzero positive-degree piece has growth dimension at most $r_ h^+$, as do all its flag bodies, including those constructed on a point blowup. For every fixed weight $u\in \mathrm{QPSH}(X)$, the integrable spaces satisfy $\dim \mathcal H_ p(u)=O(p^{r_ h^+})$.
Proof. The proposition applied to a single section excludes every positive power if $S_ h$ is empty. If all positive-degree section spaces vanish, the remaining section bounds are immediate and $\kappa (L)=-\infty $. Otherwise, suppose a graded algebra has a nonzero positive-degree piece. Write any finitely many algebraically independent elements of its ratio field with a common denominator in one positive degree, as in proposition 19.9.10. They are coordinate ratios of a section map. Its resolved image is projective algebraic by proper mapping and Chow's theorem; its dimension is therefore at least their number. The proposition bounds this dimension by $r_ h^+$, so the field's transcendence degree is at most $r_ h^+$.
Apply theorem 19.9.16 to this algebra and to the full section ring. The degree-period formula gives the growth bound in every degree, including the zero spaces outside the period. The earlier flag identification gives the body dimensions. Finally, $\mathcal H_ p(u)\subseteq H^0(X,L^ p)$, so the same upper bound holds even though the integrable spaces need not form a graded algebra. $\square$
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