Complexity one varieties are cluster type

Cluster type varieties are a class of rational varieties that generalizes toric varieties. The complexity of a log pair $(X, B)$ is an invariant that relates the dimension of $X$, the rank of the divisor class group, and the coefficients of $B$. The complexity of a Calabi-Yau pair $(X, B)$ is non- negative. If it is less than one, then X is a toric variety. Following the work of Enwright, Li, and Yáñez, we will discuss the proof of the following result: if $(X, B)$ is a Calabi-Yau pair of index one and complexity one, then X is a cluster type variety.