Communications in Analysis and Geometry
Volume 20 (2012)
Scalar curvature and uniruledness on projective manifolds
Pages: 751 – 764
It is a basic tenet in complex geometry thatnegative curvature corresponds, in a suitablesense, to the absence of rational curves on, say, a complexprojective manifold, while positive curvaturecorresponds to the abundance of rational curves. In thisspirit, we prove in this note that a projective manifold$M$ with a Kähler metric with positive total scalarcurvature is uniruled, which is equivalent to every pointof $M$ being contained in a rational curve. We also provethat if $M$ possesses a Kähler metric of total scalarcurvature equal to zero, then either $M$ is uniruled or itscanonical line bundle is torsion. The proof of the lattertheorem is partially based on the observation that if $M$is not uniruled, then the total scalar curvatures of allKähler metrics on $M$ must have the same sign, which iseither zero or negative.