Log Calabi--Yau pairs of birational complexity zero

In this article, we study the geometry of log Calabi-Yau pairs pX, Bq of index one and birational complexity zero.Firstly, we propose a conjecture that characterizes such pairs pX, Bq in terms of their dual complex and the rationality of their log canonical places.Secondly, we show that for these pairs the open set XzB is divisorially covered by open affine subvarieties which are isomorphic to open subvarieties of algebraic tori.We introduce and study invariants that measure the number of these open subvarieties of algebraic tori.Thirdly, we study boundedness properties of log Calabi-Yau pairs of index one and birational complexity zero.For instance, in dimension 2 we prove that such pairs are affinely bounded.