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arXiv · 2609.26640

Uniruledness and the sign of total scalar curvature

Zehao Sha·Jian Wang
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Abstract

For every integer $n\ge3$, we construct a smooth projective manifold $X$ of complex dimension $n$ whose canonical bundle is not pseudoeffective, or equivalently, which is uniruled, but every Kähler metric has negative total scalar curvature. In particular, $X$ admits no Kähler metric of positive scalar curvature, while it admits a Riemannian metric of positive scalar curvature. Thus the equivalence between uniruledness and the existence of a Kähler metric with positive scalar curvature holds in complex dimensions one and two, but fails in higher dimensions.

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Zehao Sha, Jian Wang. 2026-09-22. Uniruledness and the sign of total scalar curvature. https://arxiv.org/abs/2609.26640

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