## Advanced Studies in Pure Mathematics

### Another canonical compactification of the moduli space of abelian varieties

Iku Nakamura

#### 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.

#### Article information

Dates
Revised: 3 June 2009
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/ euclid.aspm/1543086129

Digital Object Identifier
doi:10.2969/aspm/05810069

Mathematical Reviews number (MathSciNet)
MR2676158

Zentralblatt MATH identifier
1213.14077

#### Citation

Nakamura, Iku. Another canonical compactification of the moduli space of abelian varieties. Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), 69--135, Mathematical Society of Japan, Tokyo, Japan, 2010. doi:10.2969/aspm/05810069. https://projecteuclid.org/euclid.aspm/1543086129