Mori dream spaces of Calabi–Yau type and the log canonicity of the Cox rings (following the paper of Yujiro Kawamata and Shinnosuke Okawa)

We prove that a Mori dream space over a field of characteristic zero is of Calabi–Yau type if and only if its Cox ring has at worst log canonical singularities. By slightly modifying the arguments we also reprove the characterization of the varieties of Fano type by the log terminality of the Cox rings.