A partial converse to the Andreotti-Grauert theorem

TL;DR

The paper proves that on smooth projective manifolds, (n-1)-ampleness of a line bundle implies (n-1)-positivity, providing a partial converse to the Andreotti-Grauert theorem, with applications to characterizing uniruled manifolds via Ricci curvature.

Contribution

It establishes a link between (n-1)-ampleness and (n-1)-positivity of line bundles, offering a partial converse to a classical theorem and applications to complex geometry.

Findings

(n-1)-ampleness implies (n-1)-positivity for line bundles.

Characterization of uniruled manifolds via Ricci curvature.

Extension of Andreotti-Grauert theorem insights.

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.