Scalar curvature and uniruledness on projective manifolds

Abstract

It is a basic tenet in complex geometry that negative curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while 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.

An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry

“…For example, in the toric case starting with a generic b ∈ t + , then {Z} is the fixed points set of the torus and so consists of a finite number of isolated points. In this case, formula (25) gives back the formulas in [31,28] as expected. Proof.…”

Scalar curvature on compact complex manifolds

“…The following conjectures are either well-known or implicitly indicated in the literatures in some special cases, and we refer to [28,5,11,6] and the references therein. Proof.…”

“…Furthermore, Yau established that a compact Kähler surface is uniruled if and only if there exists a Kähler metric with positive total scalar curvature. Recently, Heier-Wong pointed out in [11] that a projective manifold is uniruled if it admits a Kähler metric with positive total scalar curvature. By using Boucksom-Demailly-Peternell-Paun's criterion for uniruled projective manifolds ([5, Corollary 0.3]), Chiose-Rasdeaconu-Suvaina obtained in [6] a more general characterization that, a compact Moishezon manifold is uniruled if and only if it admits a smooth Gauduchon metric with positive total Chern scalar curvature.…”

“…By using Boucksom-Demailly-Peternell-Paun's criterion for uniruled projective manifolds ([5, Corollary 0.3]), Chiose-Rasdeaconu-Suvaina obtained in [6] a more general characterization that, a compact Moishezon manifold is uniruled if and only if it admits a smooth Gauduchon metric with positive total Chern scalar curvature. As motivated by these works( [28,5,11,6]), we investigate the total Chern scalar curvature of Gauduchon metrics on general compact complex manifolds. Let ω be a smooth Hermitian metric on the compact complex manifold X.…”

On projectivized vector bundles and positive holomorphic sectional curvature

“…It was proven by Kobayashi and Wu [KW70, Corollary 2] that consequently the Kodaira dimension kod(M ) = −∞. In recent work by the second named author and Wong [HW15] it has been proven that any M in H n is rationally connected, i.e., given any two points in M , there exist a connected chain of rational curves containing both points.…”