Relative spherical objects

Seidel and Thomas introduced some years ago a notion of a spherical object in the derived category $D(X)$ of a smooth projective variety $X$. Such objects induce, in a simple way, auto-equivalences of $D(X)$ called ‘spherical twists’. In a sense, they are mirror-symmetric analogues of Lagrangian spheres on a symplectic manifold and the induced auto-equivalences mirror the Dehn twists associated with the latter. We generalise this notion to the relative context by explaining what does it mean for an object of $D(Z\times X)$ to be _spherical over $Z$_ for any two separated schemes $Z$ and $X$ of finite type.
This is a joint work with Rina Anno.