## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Development of Moduli Theory — Kyoto 2013, O. Fujino, S. Kondō, A. Moriwaki, M. Saito and K. Yoshioka, eds. (Tokyo: Mathematical Society of Japan, 2016), 207 - 286

### Compactification by GIT-stability of the moduli space of abelian varieties

#### Abstract

The moduli space $\mathcal{M}_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\overline{\mathcal{M}}_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding some mildly degenerating limits of abelian varieties.

A typical case is the moduli space of Hesse cubics. Any Hesse cubic is GIT-stable in the sense that its $\mathrm{SL}(3)$-orbit is closed in the semistable locus, and conversely any GIT-stable planar cubic is one of Hesse cubics. Similarly in arbitrary dimension, the moduli space of abelian varieties is compactified by adding only GIT-stable limits of abelian varieties (§ 14).

Our moduli space is a projective “fine” moduli space of possibly degenerate abelian schemes *with non-classical non-commutative level structure* over $\mathbf{Z}[\zeta_{N},1/N]$ for some $N\geq 3$. The objects at the boundary are singular schemes, called PSQASes, projectively stable quasi-abelian schemes.

#### Article information

**Dates**

Received: 3 December 2013

Revised: 28 March 2014

First available in Project Euclid:
4 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1538622432

**Digital Object Identifier**

doi:10.2969/aspm/06910207

**Zentralblatt MATH identifier**

1373.14031

**Subjects**

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

**Keywords**

Moduli Compactification Abelian variety Heisenberg group Irreducible representation Level structure Theta function Stability

#### Citation

Nakamura, Iku. Compactification by GIT-stability of the moduli space of abelian varieties. Development of Moduli Theory — Kyoto 2013, 207--286, Mathematical Society of Japan, Tokyo, Japan, 2016. doi:10.2969/aspm/06910207. https://projecteuclid.org/euclid.aspm/1538622432