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 constant fibration with rationally connected fibres, and the base $Y$ is a canonically polarized hyperbolic projective manifold. In particular, when $Y$ is a single point, we establish that $X$ is rationally connected. Moreover, when $\dim X=3$ and $-(K_X+\Delta)$ is strictly nef, we prove that $-(K_X+\Delta)$ is ample, which confirms the singular version of a conjecture of Campana-Peternell for threefolds.