We report on joint work with Ching-Li Chai.

The *isogeny class* of a polarized abelian variety (*A,μ*) with moduli point *x* = [(*A*,μ)] is denoted by *H*(*x*), called the *Hecke orbit* of *x*. What is the Zariski closure of the set *H*(*x*)? In characteristic zero any Hecke orbit is everywhere dense in the moduli scheme of polarized abelian varieties. We easily see that this is not true in general (in the non-ordinary case) in positive characteristic. We consider and prove:

** **

**Conjecture / Theorem (HO). **

*The Hecke orbit H*([(

*A,μ*)])

*is everywhere dense in the NP*stratum given by ξ =

*N*(

*A*).

This was proved by Ching-Li Chai for *ordinary* abelian varieties (Invent. Math. 1995) and formulated as a conjecture in the general case (FO, 1995).

In order to obtain a proof, as a starting point, properties of (moduli of) *supersingular abelian varieties* were established. Several (new) concepts and their properties were needed:

• *stratifications* (EO and NP),

• *foliations *(central leaves, isogeny leaves),

• minimal *p*-divisible groups, hypersymmetric abelian varieties,

• a proof of a conjecture by Grothendieck,

• a generalization of Serre-Tate coordinates,

in order to understand the *geometry* of these moduli spaces in positive characteristic. Many aspects of these had to be established, such as dimension, irreducibility, the graph of strata with partial ordering given by ``inclusion in the boundary''. This might explain why it took us 25 years to prove the conjecture.

In this talk I will explain these notions and formulate the *basic strategy of approach*: specialization to the ``boundary'' of various strata. Examples and relevant structure theorems will be amply discussed. Then I will sketch a proof of (HO), divided into two aspects: the discrete and the continuous HO problem.