Perfect Forms and the Moduli Space of Abelian Varieties

Toroidal compactifications of the moduli space Ag, or the stack Ag, of principally polarized abelian g-folds have been constructed over C in [AMRT] and over any base in [FC]. Roughly speaking, each such compactification corresponds to choosing a way of decomposing the cone of real positive quadratic forms in g variables. The choice made here is the perfect cone decomposition, also called the first Voronoi decomposition, which leads to the compactification Ag . This carries divisor classes M , the line bundle of weight 1 modular forms, and D, the reduced boundary Ag \ Ag. (Sometimes, but not in this paper, M is denoted instead by ω.) It is easy to see that D is geometrically irreducible, and it follows ([M] or [F2]) that the classes M and D generate NS(Ag ) ⊗ Q. Here is the first main result of this paper, where a divisor class E on a projective variety X is nef if E.C ≥ 0 for all curves C on X: