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

TL;DR

This paper investigates algebraic fiber spaces with strictly nef relative anti-log canonical divisors, proving their structure and properties, including rational connectedness and hyperbolicity of the base, and confirming a conjecture for threefolds.

Contribution

It establishes the structure of such fibrations, showing they are locally constant with rationally connected fibers and hyperbolic bases, and proves the ampleness of the anti-log canonical divisor in threefolds.

Findings

Fibration is locally constant with rationally connected fibers.

Base is a canonically polarized hyperbolic manifold.

In threefolds, the strictly nef anti-log canonical divisor is ample.

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.