  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 a
reduced simply normal crossing divisor $\Delta$ on $X$ is ample if it is
numerically 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.
