Another canonical compactification of the moduli space of abelian varieties

Abstract

We construct a canonical compactification $SQ_{g,K}^{\mathrm{toric}}$ of the moduli space $A_{g,K}$ of abelian varieties over $\mathbf{Z}[\zeta_N, 1/N]$ by adding certain reduced singular varieties along the boundary of $A_{g,K}$, where $K$ is a symplectic finite abelian group, $N$ is the maximal order of elements of $K$, and $\zeta_N$ is a primitive $N$-th root of unity. In [18] a canonical compactification $SQ_{g,K}$ of $A_{g,K}$ was constructed by adding possibly non-reduced GIT-stable (Kempf-stable) degenerate abelian schemes. We prove that there is a canonical bijective finite birational morphism sq : $SQ_{g,K}^{\mathrm{toric}} \to SQ_{g,K}$. In particular, the normalizations of $SQ_{g,K}^{\mathrm{toric}}$ and $SQ_{g,K}$ are isomorphic.