Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

Abstract

Let $(X,\Delta )$ be a projective klt pair, and $f:X\to Y$ a fibration to a smooth projective variety $Y$ with strictly nef relative anti-log canonical divisor $-(K_{X/Y}+\Delta )$. We prove that $f$ is a locally trivial fibration with rationally connected fibres, and the base $Y$ is a canonically polarized hyperbolic manifold. In particular, when $Y$ is a single point, we establish that $X$ is rationally connected. Moreover, when $dimX=3$ and $-(K_X+\Delta )$ is strictly nef, we prove that $-(K_X+\Delta )$ is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.