RC-positive metrics on rationally connected manifolds

Abstract

In this paper, we prove that if a compact Kähler manifold X has a smooth Hermitian metric $\omega $ such that $(T_X,\omega )$ is uniformly RC-positive, then X is projective and rationally connected. Conversely, we show that, if a projective manifold X is rationally connected, then there exists a uniformly RC-positive complex Finsler metric on $T_X$ .

Keywords

• 53C55
• 14E08
• 14F17
• 32L20
• rationally connected
• RC-positivity
• vanishing theorem