A compactification of the moduli scheme of abelian varieties

We construct a canonical compactification $SQ^{toric}_{g,K}$ of the moduli of abelian varieties over $Z[\zeta_N, 1/N]$ where $\zeta_N$ is a primitive $N$-th root of unity. This is very similar to, but slightly diferent from the compactification constructed by the author in Inventiones vol. 136 (1999). Any degenerate abelian scheme on the boundary of $SQ^{toric}_{g,K}$ is reduced and singular and it is one of the stable quasi-abelian varieties introduced by Alexeev and the author (Tohoku J. 1999).