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

A Note on the Ampleness of Numerically Positive Log Canonical and Anti-Log Canonical Divisors