A remark on compact Kähler manifolds with nef anticanonical bundles and its applications

Abstract

Let $(X,{\omega }_X)$ be a compact Kähler manifold such that the anticanonical bundle $-K_X$ is nef. We prove that the slopes of the Harder-Narasimhan filtration of the tangent bundle with respect to a polarization of the form ${\omega }_X^{n-1}$ are semi-positive. As an application, we give a characterization of rationally connected compact Kähler manifolds with nef anticanonical bundles. As another application, we give a simple proof of the surjectivity of the Albanese map.

21 pages, Proposition 2.1 added, several references added