Advanced Studies in Pure Mathematics

Another canonical compactification of the moduli space of abelian varieties

Iku Nakamura

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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

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

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


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