Orthogonally spherical objects and spherical fibrations

We introduce a relative version of the spherical objects of Seidel and Thomas [32]. Define an object E in the derived category D(Z×X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z) and D(X) in a certain natural way. Most known examples come from subschemes of X fibred over Z. This categorifies to the notion of an object of D(Z×X) orthogonal over Z. We prove that such an object is spherical over Z if and only if it possesses certain cohomological properties similar to those in the original definition of a spherical object. We then interpret this geometrically in the case when our objects are actual flat fibrations in X over Z.