When we look at the moduli space of abelian varieties, some 'linear spaces' naturally pop up. A classical example is the Serre-Tate coordinates, which says that the local deformation space of an ordinary abelian variety is a formal torus. 'Linear subspaces' of the Serre-Tates coordinates, i.e. sub-tori, are interesting due to the connection with Shimura subvarieties. Ching-Li Chai and Frans Oort defined the notion of sustained p-divisible groups and sustained deformation spaces of abelian varieties, which provides a scheme-theoretic definition of central leaves in moduli spaces. These sustained deformation spaces are 'linear spaces' that generalize the Serre-Tate coordinates, and the strong rigidity conjecture is a conjecture that relates to their 'linear subspaces'.

We give definitions, properties, examples, and report some progress on the strong rigidity conjecture.