PomeCorollary 19.9.22.CA93F87E4E8D For every fixed holomorphic vector bundle $E$ on $X$,
\[ h^0(X,L^ p\otimes E)=O(p^ r), \qquad h^ n(X,L^ p\otimes E)=O(p^ r). \]
Corollary 19.9.22CA93F87E4E8D
Proof. For $E$ of positive rank, let $\pi :Y=\mathbb P(E^*)\to X$ parametrize lines, and let $\mathcal O_ Y(1)$ be dual tautological. Fiberwise linear forms give $\pi _*\mathcal O_ Y(1)=E$, hence
\[ H^0(X,L^ p\otimes E) =H^0(Y,(\pi ^*L)^ p\otimes \mathcal O_ Y(1)). \]
The pulled-back curvature has rank at most $r$, so the fixed-line-twist lemma, proved in all dimensions, gives the first bound. Serre duality identifies the dual of the top cohomology with
\[ H^0(X,(L^*)^ p\otimes E^*\otimes K_ X). \]
The dual curvature is $-\beta $ and has the same rank; apply the degree-zero bound to $L^*$. A bundle of rank zero has zero cohomology and satisfies both bounds. $\square$
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