Toroidal Compactification of A _g

Abstract

We first outline in general terms the various steps of our construction of the toroidal compactification of A _g , denoted by Ā _g . Precise definitions will be given in due course. These steps are:

• (a) Let X = ℤ^g. Choose a GL(X)-invariant rational polyhedral cone decomposition {σ _α } of the cone C(X) of positive semi-definite quadratic forms on X ⊗ ℝ with rational radicals such that there are only finitely many cones modulo the GL(X)-action. (Such cone decompositions are called “admissible”.)

• (b) Use the machinery of Mumford’s construction to produce a collection of semi-abelian degenerate families of principally polarized abelian varieties over complete rings, which is adapted to the combinatorial data chosen and fixed in (a).

• (c) By M. Artin’s method, approximate (instead of directly algebraizing) the complete base rings by algebraic ones to produce semi-abelian schemes over algebraic base rings. Then the theory of degeneration of polarized abelian varieties tells us that the resulting approximations can be regarded as algebraizations of those constructed in (b). The new semi-abelian schemes are still adapted to the combinatorial data chosen in (a); we will refer to them as “local models”.

• (d) The theory of degeneration allows us to glue together the local models constructed in (c) in the étale topology, because they are adapted to data (a). The result is an algebraic stack Ā _g and a semi-abelian scheme G → Ā _g which extends the universal abelian scheme over A _g .