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Compactification by GIT-stability of the moduli space of abelian varieties. (English) Zbl 1373.14031
Fujino, Osamu (ed.) et al., Development of moduli theory – Kyoto 2013. Proceedings of the 6th Mathematical Society of Japan-Seasonal Institute, MSJ-SI, Kyoto, Japan, June 11–21, 2013. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-032-7/hbk). Advanced Studies in Pure Mathematics 69, 207-286 (2016).
Summary: 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 cubits. Any Hesse cubic is GIT-stable in the sense that its 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 nonclassical noncommutative level structure over \(\mathbb{Z}[\zeta_N, 1/N]\) for some \(N\geq 3\). The objects at the boundary are singular schemes, called PSQASes, projectively stable quasi-abelian schemes.
For the entire collection see [Zbl 1353.14002].

14J10 Families, moduli, classification: algebraic theory
14K10 Algebraic moduli of abelian varieties, classification
14K25 Theta functions and abelian varieties
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