Let $X$ be a compact complex manifold, let $L \to X$ be a positive holomorphic line bundle, and let $\mathcal F$ be a coherent analytic sheaf on $X$. For each integer $m\ge 0$, write $L^m:=L^{\otimes m}$ and write $\mathcal F\otimes L^m$ for $\mathcal F\otimes_{\mathcal O_X}\mathcal O_X(L^m)$. Then there exists an integer $m_0\in\mathbb N$ such that, for every integer $m\ge m_0$ and every integer $q>0$,

\begin{align*} H^q(X,\mathcal F\otimes L^m)=0. \end{align*}