I'm currently studying toric Calabi-Yau manifolds, and in particular, am looking at how we can construct them from fans.
A fact keeps coming up in the papers I am reading, for instance in https://arxiv.org/abs/1604.07123 (page 10), that the toric variety we obtain from a fan $\Sigma$ in the lattice $N$, will be Calabi-Yau if and only if there exists some $m \in M$ such that for all $\sigma \in \Sigma (1)$, $\langle m, \sigma \rangle =1$. ($M$ is the dual to $N$).
That is, the variety is Calabi-Yau if and only if all the rays of the fan lie in a hyperplane of $N$.
I hoped to find a solution in Cox, Little and Schenck's book "Toric Varieties", but I've only managed to find a statement saying that the variety will be Gorenstein if and only if for each individual maximal cone, the rays of the cone lie in some hyperplane (this is proposition 8.2.12). This hyperplane can be different for each maximal cone. This gives us that the canonical sheaf is a line bundle, but not that it is trivial.
I'm not very at ease with canonical sheaves/canonical bundles yet, so if the answer is not too far off from this, any hints would be greatly appreciated, and an introductory paper or a set of lecture notes even more so.