A partial converse to the Andreotti-Grauert theorem

Abstract

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 $\omega$ on $X$ such that its Ricci curvature $\mathrm{Ric}(\omega)$ has at least one positive eigenvalue everywhere.