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

Abstract

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.