Nefness of the direct images of relative canonical bundles

Abstract

Given a fibration $f$ between two projective manifolds $X$ and $Y$, we discuss the nefness of the direct images $f_{\ast }(K_{X/Y}\otimes L)$, where $(L,h)$ is a pseudo-effective line bundle with mild singularity.

arXiv comment21 pages, comments are welcome! The whole article has been improved due to readers help