Complexity one varieties are cluster type

The complexity of a pair $(X,B)$ is an invariant that relates the dimension of X, the rank of the group of divisors, and the coefficients of B. If the complexity is less than one, then X is a toric variety. We prove that if the complexity is less than two, then X is a Fano type variety. Furthermore, if the complexity is less than 3/2, then X admits a Calabi--Yau structure $(X,B)$ of complexity one and index at most two, and it admits a finite cover Y → X of degree at most 2, where Y is a cluster type variety. In particular, if the complexity is one and the index is one, $(X,B)$ is cluster type. Finally, we establish a connection with the theory of T-varieties. We prove that a variety of T-complexity one admits a similar finite cover from a cluster type variety.