2012/06/12 by Gordon Heier, Heier, Gordon, Bun Wong +1 · 3 citations
Mathematics · #14J10 #14J32 #14M20 #32Q10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1206.2576
openalex publication_date 2012/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is a basic tenet in complex geometry that \it negative curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while \it positive curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifold M with a Kähler metric with positive total scalar curvature is uniruled, which is equivalent to every point of M being contained in a rational curve. We also prove that if M possesses a Kähler metric of total scalar curvature equal to zero, then either M is uniruled or its canonical line bundle is torsion. The proof of the latter theorem is partially based on the observation that if M is not uniruled, then the total scalar curvatures of all Kähler metrics on M must have the same sign, which is either zero or negative.