A partial converse to the Andreotti-Grauert theorem
英文摘要：
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 $ω$ on $X$ such that its Ricci curvature $\mathrm{Ric}(ω)$ has at least one positive eigenvalue everywhere.
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