On the log version of Serrano's conjecture

TL;DR

This paper advances the understanding of Serrano's conjecture in low dimensions by proving specific cases related to ampleness and Campana--Peternell's conjecture, with results in dimensions 3 and 4.

Contribution

It proves the ampleness conjecture for non-canonical singularities in dimension 3 and extends results on Campana--Peternell's conjecture, including cases in dimension 4.

Findings

Ampleness conjecture holds for non-canonical singularities in dimension 3.

Log canonical version of Campana--Peternell's conjecture holds in dimension 3.

In dimension 4, if -K_X is strictly nef but not ample, then certain invariants are determined, and the Picard number is constrained under specific conditions.

Abstract

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 $\rho(X)=2$.