On Extra Components in the Functorial Compactification of A _g

Abstract

Recall the following from the theory of toroidal compactifications of moduli of polarized abelian varieties (Mumford et al.[AMRT75] ℂ , Faltings and Chai[FC90]over ℤ). Denote X=ℤ^g and letCbe the convex hull in the space $Sy{{m}^{2}}(X_{\mathbb{R}}^{*})$ of semipositive symmetric matricesqwith rational null-space. For anyadmissibleGL(X)-invariant decompositionτofC(i.e. it is a face-fitting decomposition into finitely generated rational cones such that there are only finitely many cones modulo GL(X) there is a compactification $\bar{A}_{g}^{\tau }$ of the moduli spaceA _g of principally polarized abelian varieties. $\bar{A}_{g}^{\tau }$ comes with a natural stratification, and strata correspond in a 1-to-1 way to cones inτmodulo GL(X). There are infinitely many such decompositionsτand none of them seems to be better than another. True, some decompositions are smooth and projective but still there are infinitely many of these as well.