If $K_X$ is not $\mathbb Q$-Cartier then it is not nef

Question

Let $X$ be a projective variety.

Is it true that if the canonical divisor $K_X$ is not $\mathbb Q$-Cartier then it is not nef?

Answer 1

A Cartier or $\mathbf Q$-Cartier divisor $D$ on a projective variety has well-defined interesection numbers $D \cdot C$ with curves $C$. If all those intersection numbers are $\geq 0$, we say $D$ is nef. But if $D$ is not $\mathbf Q$-Cartier, there is no way in general to define $D⋅C$, and so it is not meaningful to ask if $D⋅C \geq 0$ for all curves C.