Mukai pairs and simple $K$-equivalence

A $K$-equivalent map between two smooth projective varieties is called simple if the map is resolved in both sides by single smooth blow-ups. In this paper, we will provide a structure theorem of simple $K$-equivalent maps, which reduces the study of such maps to that of special Fano manifolds. As applications of the structure theorem, we provide examples of simple $K$-equivalent maps, and classify such maps in several cases, including the case of dimension at most $8$.