On the log version of Serrano's conjecture

In this paper, we continue the study of Serrano’s conjecture in low dimensions. We focus on two special cases of the log version of Serrano’s conjecture: the ampleness conjecture and the log version of Campana–Peternell’s conjecture. In dimension 3, we prove that the ampleness conjecture holds for non-canonical singularities; by the same method, we also prove that the log canonical version of Campana–Peternell’s conjecture holds in dimension 3. In dimension 4, we improve the results on Campana–Peternell’s conjecture by excluding the case that the numerical dimension of the anti-canonical divisor is 3. Specifically, we show that for a projective smooth fourfold $X$, if $-K_X$ is strictly nef but not ample, then $\kappa (X,-K_X)=0$ and $\nu (X,-K_X)=2$; in this case, if we further assume that $X$ admits a Fano contraction $X\to Y$ onto a surface $Y$ induced by some extremal ray, then $\rho (X)=2$.