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.

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• Logarithmic vanishing theorems for effective $q$-ample divisors
• Strictly nef vector bundles and characterizations of ${\mathbb{P}}^n$
• Integral inequalities for holomorphic maps and applications
• RC-positivity and scalar-flat metrics on ruled manifolds