Harvard-MIT Algebraic Geometry Seminar
		      March 5, 2002 at 3:00 p.m.
			   Harvard Room 507



		  Moduli spaces of abelian varieties
		      in positive characteristic



	      FRANS OORT (Universiteit Utrecht and MIT)



Abstract:

Abelian varieties in characteristic p have additional structures (the
action of Frobenius on the p-divisible group, Newton polygons,
structures of the p-kernel) which give access to geometric aspects of
these moduli spaces in all characteristics.

Here is one motivating

CONJECTURE. For g>2 the moduli space of abelian varieties of dimension
g over the complex numbers does not contain a compact codimension g
subvariety.

In this talk we discuss various structures, constructions, theorems,
conjectures, and we indicate interrelations between the following
aspects:

(GC) A conjecture by Grothendieck on the realization of Newton
Polygons in families of p-divisible groups (Grothendieck, Montreal
1970, now a theorem). One essential ingredient for this proof is the
"Purity theorem" (joint work with Johan de Jong). As a corollary: a new
proof of the conjecture by Manin that every symmetric NP is realized
by an abelian variety.

(NP) The stratification by Newton Polygons of the moduli space of
abelian varieties, and of deformation spaces of p-divisible groups. We
compute the dimension of these strata.

(EO) The stratification by the structure of p-kernels (joint work with
Torsten Ekedahl). This stratification is now widely studied in various
disguises.  As a corollary: a new proof of the irreducibility of the
moduli space of principally polarized abelian varieties in arbitrary
characteristic.

(Fol) Two different "foliations" of NP-strata. Conjecturally these
describe the Zariski-closure of Hecke orbits in positive
characteristic.  These structures give access to a precise description
of an "almost product-structure" of these leaves covering an open NP
stratum.  This structure is used by Elena Mantovan in a computation of
the cohomology of certain Shimura varieties of PEL type.

(p-Div) Possibly we will discuss new results by Thomas Zink on
families of p-divisible groups.