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A partial converse to the Andreotti–Grauert theorem
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 1 / January 2019
- Published online by Cambridge University Press:
- 19 November 2018, pp. 89-99
- Print publication:
- January 2019
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- Article
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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.
