## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), I. Nakamura and L. Weng, eds. (Tokyo: Mathematical Society of Japan, 2010), 69 - 135

### 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.

#### Article information

**Dates**

Received: 31 March 2009

Revised: 3 June 2009

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/05810069

**Mathematical Reviews number (MathSciNet)**

MR2676158

**Zentralblatt MATH identifier**

1213.14077

**Subjects**

Primary: 14J10: Families, moduli, classification: algebraic theory 14K10: Algebraic moduli, classification [See also 11G15] 14K25: Theta functions [See also 14H42]

#### 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