Let $B$ a smooth projective connected variety over $\mathbf C$.
Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero.
Does the converse hold? That is, suppose that $B$ is of Kodaira dimension zero. Does it follow that $K_B$ is torsion?
This is true when $\dim B\leq 2$, but I don't know whether this is true when $\dim B>2$.
If not true when $\dim B>2$, what non-trivial properties can we show $K_B$ to have if $X$ is of Kodaira dimension zero?