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Logarithmic abelian varieties, Part IV: Proper models

Part of: Abelian varieties and schemes Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  11 January 2016

Takeshi Kajiwara ,
Kazuya Kato  and
Chikara Nakayama
Takeshi Kajiwara
Affiliation:
Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya-ku, Yokohama 240-8501, Japan, [email protected]
Kazuya Kato
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637, USA, [email protected]
Chikara Nakayama
Affiliation:
Department of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi Tokyo, 186-8601, Japan, [email protected]
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Abstract

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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.

MSC classification

Secondary: 14K10: Algebraic moduli, classification 14J10: Families, moduli, classification 14D06: Fibrations, degenerations
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 9 - 63
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Artin, M., “Algebraization of formal moduli, I” in Global Analysis (Papers in Honor of K. Kodaira), University of Tokyo Press, Tokyo, 1969, 21–71. MR 0260746.Google Scholar
[2] Artin, M., “The implicit function theorem in algebraic geometry” in Algebraic Geometry (Bombay, 1968), Oxford University Press, London, 1969, 13–34. MR 0262237.Google Scholar
[3] de Jong, A. J. et al., “Morphisms of algebraic spaces” in The Stacks Project, Chapter 51, May 29, 2015, http://stacks.math.columbia.edu/download/spaces-morphisms.pdf. Google Scholar
[4] Faltings, G. and Chai, C., Degeneration of Abelian Varieties, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990. MR 1083353. DOI 10.1007/978-3-662-02632-8.Google Scholar
[5] Grothendieck, A., with Artin, M. and Verdier, J.-L., Théorie des topos et cohomologie étale des schémas (SGA4), Lecture Notes in Math. 269, Springer, Berlin, 1972. MR 0354652. Google Scholar
[6] Grothendieck, A., Étude globale élémentaire de quelques classes de morphismes (EGA II), Publ. Math. Inst. Hautes Études Sci. 8 (1961). MR 0163909.Google Scholar
[7] Illusie, L., “Logarithmic spaces (according to K. Kato)” in Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math. 15, Academic Press, San Diego, Calif., 1994, 183–203. MR 1307397. CrossRefGoogle Scholar
[8] Kajiwara, T., Kato, K., and Nakayama, C., Logarithmic abelian varieties, I: Complex analytic theory, J. Math. Sci. Univ. Tokyo 15 (2008), 69–193. MR 2422590.Google Scholar
[9] Kajiwara, T., Kato, K., and Nakayama, C., Logarithmic abelian varieties, II: Algebraic theory, Nagoya Math. J. 189 (2008), 63–138. MR 2396584.CrossRefGoogle Scholar
[10] Kajiwara, T., Kato, K., and Nakayama, C., Logarithmic abelian varieties, III: Logarithmic elliptic curves and modular curves, Nagoya Math. J. 210 (2013), 59–81. MR 3079275. DOI 10.1215/00277630-2077017.Google Scholar
[11] Kato, K., Toric singularities, Amer. J. Math. 116 (1994), 1073–1099. MR 1296725. DOI 10.2307/2374941.CrossRefGoogle Scholar
[12] Kajiwara, T., Kato, K., and Nakayama, C., Logarithmic structures of Fontaine–Illusie, II: Logarithmic flat topology, preprint, 1991.Google Scholar
[13] Kajiwara, T., Kato, K., and Nakayama, C., Logarithmic Dieudonné theory, preprint, 1992.Google Scholar
[14] Kato, K. and Usui, S., Classifying Spaces of Degenerating Polarized Hodge Structures, Ann. of Math. Stud. 169, Princeton University Press, Princeton, 2009. MR 2465224. Google Scholar
[15] Knutson, D., Algebraic Spaces, Lecture Notes in Math. 203, Springer, Berlin, 1971. MR 0302647. Google Scholar
[16] Laumon, G. and Moret-Bailly, L., Champs algébriques, Ergeb. Math, Grenzgeb. (3) 39, Springer, Berlin, 2000. MR 1771927. Google Scholar
[17] Nakayama, C., Logarithmic étale cohomology, Math. Ann. 308 (1997), 365–404. MR 1457738. DOI 10.1007/s002080050081.Google Scholar
[18] Nizioł, W., K-theory of log-schemes, I, Doc. Math. 13 (2008), 505–551. MR 2452875. Google Scholar
[19] Olsson, M. C., Log algebraic stacks and moduli of log schemes, Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2001. MR 2702292. Google Scholar
[20] Raynaud, M., “Variétés abéliennes et géométrie rigide” in Actes du Congrès International des Mathématiciens, I (Nice, 1970), Gauthier-Villars, Paris, 1971, 473–477. MR 0427326.Google Scholar