Families of surfaces with many CM fibres

It is well known that the moduli spaces of abelian varieties and K3 surfaces are (essentially) Shimura varieties. This implies that any such variety can be deformed to one that is of CM type, and in this deformation we can even require that some given collection of Hodge classes is preserved. My talk is based on the question how much we can say about such things once we leave the world of Shimura varieties. Already for surfaces of general type this brings us into unexplored terrain. In my talk I will explain how to obtain many families of surfaces in which the CM fibres lie dense.