On morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature

In this paper, with the aim of establishing a structure theorem for a compact Kähler manifold X with semi-positive holomorphic sectional curvature, we study a morphism ϕ: X Y to a compact Kähler manifold Y with pseudo-effective canonical bundle. We prove that the morphism ϕ is always smooth (that is, a submersion), the image Y admits a finite etale cover T Y by a complex torus T, and further that all the fibers are isomorphic when X is projective. Moreover, by applying a modified method to maximal rationally connected fibrations, we show that X is rationally connected, if X is projective and X has no truly flat tangent vectors at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive). This result gives a generalization of Yau's conjecture. As a further application, we obtain a uniformization theorem for compact Kähler surfaces with semi-positive holomorphic sectional curvature.