Abstract. We develop the algebraic theory of log abelian varieties. This is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ( [6]), where we developed the analytic theory of log abelian varieties. Part II. Algebraic theory IntroductionThis is Part II of our series of papers on log abelian varieties, and is an algebraic counterpart of the previous Part I ( [6]), where we developed the analytic theory of log abelian varieties.Degenerating abelian varieties can not preserve group structures, properness, and smoothness at the same time. However, in a log world, degenerating abelian varieties can become group objects called log abelian varieties which behave like proper smooth objects. For some background, see the introduction of Part I.In Part I [6], we studied a complex analytic theory of log abelian varieties. In this Part II and sequel, we develop an algebraic theory of log abelian varieties.Our philosophy, main ideas, and methods are illustrated in the introductory section, Section 1, by using Tate elliptic curves as examples. The main definitions and basic results concerning log abelian varieties are given in Sections 2-4. In Section 2 we introduce the notion log 1-motif, and by using it, we define in Section 3 log abelian variety with constant degeneration which is a special case of log abelian variety. We define log abelian variety in Section 4. The proofs of the results are given in Sections 5-11. In Sections 1-4, we refer to Sections 5-11 for almost all the proofs. (In the beginning of each section 1-4, we give a list of propositions whose proofs are put off and explain where we give their proofs.) The reason why we put the proofs in later sections is that the proofs of some results are very long and we wish that the reader can understand the main story early. A reader who prefers to know the proofs right after the results appear could read this paper in the following order: 1, 5, 2, 6, 3, 7, 8, 4 and 9-11. In the sequel (Part III, etc.), one of the main subjects will be moduli spaces of log abelian varieties.While we were writing this paper, we learned that V. Pahnke completed a beautiful work [12] on log abelian varieties. His formulation is different from ours: He works with the inverse limits of blowing ups along log structures. Log elliptic curves were already studied by M. C. Olsson in [11]. He also worked with the inverse limits of blowing ups along log structures.Acknowledgments. A part of this work was done while the second author was a visitor of University of Minnesota in the spring of 1992. He expresses his sincere gratitudes, especially to Professor W. Messing, for the hospitality and stimulating discussions. The authors are very much grateful to Professor Kazuhiro Fujiwara for enlightening discussions. They learned much from his unpublished work [3] on degeneration of abelian varieties. §1. Tate curvesIn this section, we explain our methods by using Tate curves as examples. The proof of Proposition 1.7 is given in Section 5.1.1...
This is Part VI of our series of papers on log abelian varieties. In this part, we study local moduli and GAGF of log abelian varieties.Contents §1. GAGF for G m -, G m,log -, and G m,log /G m -torsors on weak log abelian varieties §2. Két presentation of a weak log abelian variety by a model, and equivalences with the categories of models §3. Moduli in the case of constant degeneration §4. Weak log abelian varieties over complete discrete valuation rings, I §5. GAGF for log abelian varieties, I §6. GAGF for log abelian varieties, II §7. Weak log abelian varieties over complete discrete valuation rings, II Primary 14K10; Secondary 14J10, 14D06The results in Sections 1 and 2 themselves are important and used in the proof of the GAGF. Section 5 proves the part of the full faithfulness of the GAGF. The theoretical dependence of the sections are as follows.2 ⇒ 5. 1, 2, 3, 4, 5 ⇒ 6 ⇒ 7.
This is part IV of our series of articles on log abelian varieties. In this part, we study the algebraic theory of proper models of log abelian varieties.
Abstract. We introduce a log Picard variety over the complex number field by the method of log geometry in the sense of Fontaine-Illusie, and study its basic properties, especially, its relationship with the group of log version of Gm-torsors.
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