Nagoya Mathematical Journal

Analytic log Picard varieties

Takeshi Kajiwara, Kazuya Kato, and Chikara Nakayama

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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 $\mathbb{G}_{m}$-torsors.

Article information

Nagoya Math. J., Volume 191 (2008), 149-180.

First available in Project Euclid: 17 September 2008

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Zentralblatt MATH identifier

Primary: 14K30: Picard schemes, higher Jacobians [See also 14H40, 32G20]
Secondary: 14K20: Analytic theory; abelian integrals and differentials 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]


Kajiwara, Takeshi; Kato, Kazuya; Nakayama, Chikara. Analytic log Picard varieties. Nagoya Math. J. 191 (2008), 149--180.

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  • L. Breen, Extensions du groupe additif, Publ. Math., Inst. Hautes Étud. Sci., 48 (1977), 39--125.
  • T. Fujisawa, Limits of Hodge structures in several variables, Compositio Math., 115 (1999), 129--183.
  • T. Fujisawa and C. Nakayama, Mixed Hodge structures on log deformations, Rendiconti del Seminario Matematico di Padova, 110 (2003), 221--268.
  • L. Illusie, Complexe cotangent et déformations I, II, Lect. Notes Math. 239, 283, Berlin-Heidelberg-New York, Springer, 1972.
  • L. Illusie, K. Kato, and C. Nakayama, Quasi-unipotent logarithmic Riemann-Hilbert correspondences, J. Math. Sci. Univ. Tokyo, 12 (2005), 1--66.
  • T. Kajiwara, Logarithmic compactifications of the generalized Jacobian variety, J. Fac. Sci. Univ. Tokyo Sect. IA, Math., 40 (1993), 473--502.
  • T. Kajiwara, Log jacobian varieties, I: Local theory, in preparation.
  • T. Kajiwara, K. Kato, and C. Nakayama, Logarithmic abelian varieties, Part I: Complex analytic theory, J. Math. Sci. Univ. Tokyo, 15 (2008), 69--193.
  • T. Kajiwara, K. Kato, and C. Nakayama, Logarithmic abelian varieties, Part II. Algebraic theory, Nagoya Math. J., 189 (2008), 63--138.
  • T. Kajiwara and C. Nakayama, Higher direct images of local systems in log Betti cohomology, preprint, submitted.
  • M. Kashiwara, A study of variation of mixed Hodge structure, Publ. Res. Inst. Math. Sci., Kyoto Univ., 22 (1986), 991--1024.
  • K. Kato, T. Matsubara, and C. Nakayama, Log $C^\infty$-functions and degenerations of Hodge structures, Advanced Studies in Pure Mathematics 36, Algebraic Geometry 2000, Azumino (S. Usui, M. Green, L. Illusie, K. Kato, E. Looijenga, S. Mukai, and S. Saito, eds.), 2002, pp. 269--320.
  • K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over $\bold C$, Kodai Math. J., 22 (1999), 161--186.
  • K. Kato and S. Usui, Classifying spaces of degenerating polarized Hodge structures, to appear in Ann. of Math. Studies, Princeton Univ. Press.
  • K. Kato, C. Nakayama, and S. Usui, $SL(2)$-orbit theorem for degeneration of mixed Hodge structure, J. Algebraic Geometry, 17 (2008), 401--479.
  • Y. Kawamata, On algebraic fiber spaces, Contemporary trends in algebraic geometry and algebraic topology (Shiing-Shen Chern, Lei Fu, and Richard Hain, eds.), Nankai Tracts in Mathematics, vol. 5, World Scientific Publishing, 2002, pp. 135--154.
  • Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math., 118 (1994), 395--409.
  • G. Kempf, F. Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings, I, Lect. Notes Math. 339, 1973.
  • S. MacLane, Homologie des anneaux et des modules, C\.B\.R\.M. Louvain (1956), 55--80.
  • Y. Namikawa, Toroidal degeneration of abelian varieties, II, Math. Ann., 245 (1979), 117--150.
  • M. Saito, Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ., 24 (1988), 849--995.
  • J. H. M. Steenbrink, Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures, Math. Ann., 301 (1995), 105--118.
  • J. H. M. Steenbrink and S. Zucker, Variation of mixed Hodge structure. I, Invent. Math., 80 (1985), 489--542.
  • I. Vidal, Monodromie locale et fonctions Zêta des log schémas, Geometric aspects of Dwork Theory, volume II (A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, and F. Loeser, eds.), Walter de Gruyter, Berlin, New York, 2004, pp. 983--1039.