Derived categories of cyclic covers and their branch divisors

Given a variety $Y$ with a rectangular Lefschetz decomposition of its derived category, I will discuss an interesting relation between the derived categories of a cyclic cover of $Y$ and its branch divisor. As examples, I will describe the cases of cyclic cubic fourfolds and "special Gushel--Mukai varieties," where we get a description of the equivariant category of certain K3 categories. This is joint work with Alexander Kuznetsov.