Some non-vanishing results on log canonical pairs of dimension 4

Abstract

Let $(X,Δ)$ be a log canonical pair over $\mathbb{C}$ with $X$ a normal projective variety, $Δ$ an effective $\mathbb{Q}$-divisor, and $K_X+Δ$ nef. We give a non-vanishing criterion for $K_X+Δ$ in dimension $n$ with $X$ uniruled, assuming various conjectures of LMMP in dimensions (up to) $n-1$ or $n$, and a semi-ampleness criterion in the irregular case. In particular, we obtain that if $X$ is a uniruled $4$-fold, then $κ(K_X+Δ)\geq 0$ and if $X$ is a $4$-fold with $q(X)>0$, then $K_X+Δ$ is semi-ample.