Moduli of abelian varieties in mixed and in positive characteristic

We start with a discussion of CM abelian varieties in characteristic zero, and in positive characteristic. An abelian variety over a finite field is a CM abelian variety, as Tate proved. Can it be CM lifted to characteristic zero? Here are other questions. Does there exist an abelian variety, say over Qa, or over Fp, of dimension g > 3 not isogenous with the Jacobian of an algebraic curve? Can we construct algebraic curves, say over C, where the Jacobian is a CM abelian variety? We give (partial) answers to these questions and discuss stratifications and foliations of moduli spaces of abelian varieties in positive characteristic.