Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten. (Arithmetical compactification of the moduli space of abelian varieties). (German) Zbl 0597.14036

Faltings, Gerd

Arbeitstag. Bonn. 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 321-383 (1985).

[For the entire collection see Zbl 0547.00007.]
The main object of this article is to construct compactifications of the moduli space of principally polarized abelian varieties of dimension g over Spec(\({\mathbb{Z}})\). The existence of such compactification for each fibre (i.e. Spec(\({\mathbb{Q}})\) etc.) has been known since long. The author succeeded to globalize it in showing that one can apply Mumford’s method of toroidal compactification (over \({\mathbb{C}})\) also in this case by connecting it with another Mumford’s theory [D. Mumford, Compos. Math. 24, 239-272 (1972; Zbl 0241.14020)].
The compactifications are realized as algebraic stacks, but there are finite coverings (over Spec(\({\mathbb{Z}}[1/n]))\) which are algebraic spaces.
As applications, one can simplify some arguments in the proof of Mordell conjecture by the same author [Invent. Math. 73, 349-366 (1983; Zbl 0588.14026)], and one can show the irreducibility of the moduli space of principally polarized abelian varieties over any characteristic (known before only for characteristic 0 or for low dimension).

Reviewer: Y.Namikawa

MSC:

14K10	Algebraic moduli of abelian varieties, classification
14D15	Formal methods and deformations in algebraic geometry
14K15	Arithmetic ground fields for abelian varieties
14D20	Algebraic moduli problems, moduli of vector bundles

Keywords:

semi-abelian variety; compactifications of the moduli space of principally polarized; abelian varieties; toroidal compactification; proof of Mordell conjecture; compactifications of the moduli space of principally polarized abelian varieties

Citations:

Zbl 0547.00007; Zbl 0241.14020; Zbl 0588.14026