Strongly HN-Positivity, Uniformly RC k-Positivity and Rational Connectedness

Abstract

In this paper, we discuss the concept of HN-semipositive (HN-positive) vector bundle and also introduce strongly HN-semipositive vector bundle over compact complex manifold. Let M be a projective manifold with HN-semipositive tangent bundle. If M is rationally connected, we show that T^1,0M is strongly HN-positive. We give a characterization of rationally connected compact Kähler manifolds with strongly HN-semipositive tangent bundle. In the second part, we show that a uniformly RC k-positivity implies mean curvature positivity.