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Let $(X,\Delta)$ be a projective klt pair, and $f:X\to Y$ a fibration to asmooth projective variety $Y$ with strictly nef relative anti-log canonicaldivisor $-(K_{X/Y}+\Delta)$. We prove that $f$ is a locally constant fibrationwith rationally connected fibres, and the base $Y$ is a canonically polarizedhyperbolic projective manifold. In particular, when $Y$ is a single point, weestablish 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, whichconfirms the singular version of a conjecture of Campana-Peternell forthreefolds.

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