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

Abstract

Let (Formula presented.) be a projective klt pair, and (Formula presented.) a fibration to a smooth projective variety (Formula presented.) with strictly nef relative anti-log canonical divisor (Formula presented.). We prove that (Formula presented.) is a locally trivial fibration with rationally connected fibres, and the base (Formula presented.) is a canonically polarized hyperbolic manifold. In particular, when (Formula presented.) is a single point, we establish that (Formula presented.) is rationally connected. Moreover, when (Formula presented.) and (Formula presented.) is strictly nef, we prove that (Formula presented.) is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds. © 2024 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.