Some non-vanishing results on log canonical pairs of dimension 4
Let $(X,\Delta)$ be a log canonical pair over $\mathbb{C}$ with $X$ a normal projective variety, $\Delta$ an effective $\mathbb{Q}$-divisor, and $K_X+\Delta$ nef. We give a non-vanishing criterion for $K_X+\Delta$ 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 $\kappa(K_X+\Delta)\geq 0$ and if $X$ is a $4$-fold with $q(X)>0$, then $K_X+\Delta$ is semi-ample.