A partial converse to the Andreotti–Grauert theorem

Let $X$ be a smooth projective manifold with $\dim _{\mathbb{C}}X=n$ . We show that if a line bundle $L$ is $(n-1)$ -ample, then it is $(n-1)$ -positive. This is a partial converse to the Andreotti–Grauert theorem. As an application, we show that a projective manifold $X$ is uniruled if and only if there exists a Hermitian metric $\unicode[STIX]{x1D714}$ on $X$ such that its Ricci curvature $\text{Ric}(\unicode[STIX]{x1D714})$ has at least one positive eigenvalue everywhere.