A Remark on Compact Kähler Manifolds with Nef Anticanonical Bundles and Its Applications

Let (X, ωX) be a compact Kähler manifold such that the anticanonical bundle −KX is nef. We prove that the slopes of the HarderNarasimhan filtration of the tangent bundle with respect to a polarization of the form ω X 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. 0. Introduction Compact Kähler manifolds with semipositive anticanonical bundles have been studied in depth in [CDP12], where a rather general structure theorem for this type of manifolds has been obtained. It is a natural question to find some similar structure theorems for compact Kähler manifolds with nef anticanonical bundles. Obviously, we cannot hope the same structure theorem for this type of manifolds (cf. [CDP12, Remark 1.7]). It is conjectured that the Albanese map is a submersion and that the fibers exhibit no variation of their complex structure (cf. [CH13] for some special cases). In relation with the structure of compact Kähler manifolds with nef anticanonical bundles, it is conjectured in [Pet12, Conj. 1.3] that the tangent bundles of projective manifolds with nef anticanonical bundles are generically nef. We first recall the notion of generically semipositive (resp. strictly positive) (cf. [Miy87, Section 6]) Definition 0.1. Let X be a compact Kähler manifold and let E be a vector bundle on X. Let ω1, · · · , ωn−1 be Kähler classes. Let 0 ⊂ E0 ⊂ E1 ⊂ · · · ⊂ Es = E (resp. Ω 1 X) be the Harder-Narasimhan semistable filtration with respect to (ω1, · · · , ωn−1). We say that E is generically (ω1, · · · , ωn−1)-semipositive (resp. strictly positive), if ∫ X c1(Ei+1/Ei) ∧ ω1 ∧ · · · ∧ ωn−1 ≥ 0 (resp. > 0) for all i. If ω1 = · · · = ωn−1, we write the polarization as ω n−1 1 for simplicity. We rephrase [Pet12, Conj. 1.3] as follows Conjecture 0.1. Let X be a projective manifold with nef anticanonical bundle. Then TX is generically (H1, · · · , Hn−1)-semipositive for any (n− 1)-tuple of ample divisors H1, · · · , Hn−1. In this article, we first give a partial positive answer to this conjecture. More precisely, we prove 1 ha l-0 08 23 94 7, v er si on 1 19 M ay 2 01 3