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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
##### Keywords:
log geometry; log abelian varieties
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##### References:
 [1] P. Deligne and M. Rapoport, Les schemas de modules de courbes elliptiques , Modular functions of one variable, II, Antwerp 1972 Preceedings (P. Deligne and W. Kuyk, eds.), Lecture Notes in Math. 349, Berlin, Springer-Verlag, 1973, pp. 143–316. · Zbl 0281.14010 [2] G. Faltings and C. Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge$$\cdot$$Band 22, Springer-Verlag, Berlin, 1990. · Zbl 0744.14031 [3] K. Fujiwara, Arithmetic compactifications of Shimura varieties $$(I)$$ , preprint (1990). [4] A. Grothendieck and J. A. Dieudonne, Étude cohomologique des faisceaux cohérents (EGA III), Publ. Math., Inst. Hautes Étud. Sci. 11 (1961), 17 (1963). [5] L. Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology , Cohomologies $$p$$-adiques et applications Arithmétiques (II) (P. Berthelot, J. M. Fontaine, L. Illusie, K. Kato and M. Rapoport, éd.), Astérisque 279, 2002, pp. 271–322. [6] T. Kajiwara, K. Kato, and C. Nakayama, Logarithmic abelian varieties, Part I: Complex analytic theory, to appear in J. Math. Sci. Univ. Tokyo. · Zbl 1156.14038 [7] K. Kato, Logarithmic structures of Fontaine-Illusie , Algebraic analysis, geometry, and number theory (J.-I. Igusa, ed.), Johns Hopkins University Press, Baltimore, 1989, pp. 191–224. · Zbl 0776.14004 [8] K. Kato, Toric singularities , Amer. J. Math., 116 (1994), 1073–1099. JSTOR: · Zbl 0832.14002 · doi:10.2307/2374941 · links.jstor.org [9] K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $$\mathbbC$$ , Kodai Math. J., 22 (1999), 161–186. · Zbl 0957.14015 · doi:10.2996/kmj/1138044041 [10] C. Nakayama, Logarithmic étale cohomology , Math. Ann., 308 (1997), 365–404. · Zbl 0877.14016 · doi:10.1007/s002080050081 [11] M. C. Olsson, Log algebraic stacks and moduli of log schemes , · JFM 49.0311.01 [12] V. Pahnke, Uniformisierung log-abelscher Varietäten , Doctor thesis, Universität Ulm (2005). · Zbl 1115.14039 [13] P. Roquette, Analytic theory of elliptic functions over local fields, Hamb. Math. Einzelschriften. Neue Folge$$\cdot$$Heft 1, Vandenhoeck & Ruprecht in Göttingen, 1970. · Zbl 0194.52002 [14] J. H. Silverman, The arithmetic of elliptic curves, Graduate texts in mathematics; 106, Springer-Verlag, 1986. · Zbl 0585.14026
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