Birationally rigid iterated Fano double covers

Iterating the procedure of making a double cover over a given variety, we construct large families of smooth higher-dimensional Fano varieties of index 1. These varieties can be realized as complete intersections in various weighted projective spaces. We prove that a generic variety of these families is birationally superrigid. In particular, it admits no non-trivial structure of a fibration into rationally connected (or even uniruled) varieties, it is non-rational, and its groups of birational and biregular self-maps coincide.