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

Abstract

A. Let ( , ) be a projective klt pair, and ∶ → a fibration to a smooth projective variety with strictly nef relative anti-log canonical divisor −( ∕ + ). We prove that is a locally constant fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic projective manifold. In particular, when is a single point, we establish that is rationally connected. Moreover, when dim = 3 and −( + ) is strictly nef, we prove that −( + ) is ample, which confirms the singular version of the Campana-Peternell conjecture for threefolds.C * (− ( ∕ + ) + )and *

The Nonvanishing Problem for varieties with nef anticanonical bundle

“…if K X intersects every curve on X positively, then it is expected that X is a Fano variety. This was confirmed in dimension 3 in [63,73] and in many cases in dimension 4 in [62]. When X is a smooth, projective, rationally connected fourfold with K X strictly nef, then Ä.X; K X / 0 by [62].…”