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Logarithmic abelian varieties. (English) Zbl 1169.14031
In a series of papers the authors study degenerations of abelian varieties in the context of log geometry in the sense of Fontaine and Illusie. As is well known, degenerating abelian varieties cannot preserve the group structure, properness and smoothness at the same time. However as log objects the denenerations become group objects, called log abelian varieties. They behave similarly as usual abelian varieties. The first part of this series [J. Math. Sci. Tokyo 15, 69–193 (2008; Zbl 1156.14038] contains the analytic theory of log abelian varieties. This second part is an algebraic counterpart of the first one and develops an algebraic theory of log abelian varieties. Section 1 illustrates the main ideas and methods using Tate’s elliptic curves as examples. Sections 2 to 4 contain the main definitions and basic results: log 1-motifs are introduced, and log abelian varieties are defined. Finally, proofs of the results are given in the remaining sections. In a third forthcoming part the authors promise to study moduli spaces of log abelian varieties.

MSC:
14K10 Algebraic moduli of abelian varieties, classification
14J10 Families, moduli, classification: algebraic theory
14D06 Fibrations, degenerations in algebraic geometry
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