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

Abstract

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 $\varphi :X\to Y$ to a compact Kähler manifold $Y$ with pseudo-effective canonical bundle. We prove that the morphism $\varphi$ is always smooth (that is, a submersion), the image $Y$ admits a finite etale cover $T\to Y$ 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.

29 pages, comments are welcome

References in corpus (2)

• RC-positive metrics on rationally connected manifolds
• On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature

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• Positive curvature operator, projective manifold and rational connectedness
• Deformation of Hermitian metrics