PomeLemma 19.9.21.FCAD080F93F6 For every fixed holomorphic line bundle $A$ on $X$,
\[ h^0(X,L^ p\otimes A)=O(p^ r). \]
Lemma 19.9.21FCAD080F93F6
Proof. Induct on the dimension, simultaneously for all compact complex manifolds and all integer upper bounds on curvature rank. Dimension zero is immediate. If all these section spaces vanish, there is nothing to prove. Otherwise fix a nonzero section in degree $a\geq 1$ and resolve its zero divisor on a smooth compact modification $\mu :Y\to X$. With $M=\mu ^*L$, the pulled-back twist is $M^{-a}(D)$ for a fixed effective divisor $D$ whose components are smooth. Pullback is injective on sections, and the pulled-back curvature has rank at most $r$.
Write $D=H_1+\cdots +H_ s$, allowing repeated components, and put $D_ j=H_1+\cdots +H_ j$. For $m=p-a>0$, the divisor sequences give
\[ h^0(Y,M^ m(D)) \leq h^0(Y,M^ m)+\sum _{j=1}^ s h^0\bigl (H_ j,(M|_{H_ j})^ m\otimes \mathcal O_ Y(D_ j)|_{H_ j}\bigr ). \]
The first term is $O(m^ r)$ by the preceding corollary; every other term is $O(m^ r)$ by induction, since curvature rank cannot increase on restriction. The number of terms and their twists are fixed. Absorb the finitely many $p\leq a$ in the constant. On disconnected manifolds apply this argument componentwise; this also makes the induction valid for disconnected divisor components. $\square$
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