A note on the ampleness of numerically positive log canonical and anti-log canonical divisors

In this short note, we consider the conjecture that the log canonical divisor (resp. the anti-log canonical divisor) $K_X+\mathrm{\Delta }$ (resp. $-(K_X+\mathrm{\Delta })$) on a pair $(X,\mathrm{\Delta })$ consisting of a complex projective manifold $X$ and a reduced simply normal crossing divisor $\mathrm{\Delta }$ on $X$ is ample if it is numerically positive. More precisely, we prove the conjecture for $K_X + \Delta$with$\Delta \neq 0$indimension4andfor$-(K_X + \Delta)$with$\Delta \neq 0$ in dimension 3 or 4.