Kato, Fumiharu Log smooth deformation theory. (English) Zbl 0876.14007 Tôhoku Math. J., II. Ser. 48, No. 3, 317-354 (1996). This paper lays a foundation for log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor on it, and (2) global smoothings of normal crossing varieties.The former is a generalization of the relative deformation theory introduced by Makio and others, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa. Reviewer: F.Kato (Mannheim) Cited in 1 ReviewCited in 58 Documents MSC: 14D15 Formal methods and deformations in algebraic geometry 13D10 Deformations and infinitesimal methods in commutative ring theory 14M25 Toric varieties, Newton polyhedra, Okounkov bodies Keywords:log smooth deformation functor; relative deformations; global smoothings × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] R. FRIEDMAN, Global smoothings of varieties with normal crossings, Ann. of Math. (2) 118 (1983), 75-114. JSTOR: · Zbl 0569.14002 · doi:10.2307/2006955 [2] A. GROTHENDIECK, Revetements etales et groupe fondamentale, Lecture Notes in Math. 224, Springer-Verlag, Berlin, 1971. · Zbl 0222.14002 [3] L. ILLUSIE, Introduction a la geometric logarithmique, Seminar notes at Univ. of Tokyo in 1992 [4] T. KAJIWARA, Logarithmic compactifications of the generalized Jacobian variety, J. Fac. Sci. Univ Tokyo, Sect. IA, Math. 40 (1993), 473-502. · Zbl 0817.14011 [5] K. KATO, Logarithmic structures of Fontaine-IIIusie, in Algebraic Analysis, Geometry and Numbe Theory (J. -I. Igusa, ed.), Johns Hopkins Univ., 1988, 191-224. · Zbl 0776.14004 [6] Y. KAWAMATA AND Y. NAMIKAWA, Logarithmic deformations of normal crossing varieties an smoothings of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), 395^09. · Zbl 0848.14004 · doi:10.1007/BF01231538 [7] K. KODAIRA AND D. C. SPENCER, On deformation of complex analytic structures, I-II, Ann. of Math (2) 67 (1958), 328-466. JSTOR: · Zbl 0128.16901 · doi:10.2307/1970009 [8] S. LICHTENBAUM AND M. ScHLESSiNGER, The cotangent complex of a morphism, Trans. Amer. Math Soc. 128 (1967), 41-70. JSTOR: · Zbl 0156.27201 · doi:10.2307/1994516 [9] K. MAKIO, On the relative pseudo-rigidity, Proc. Japan Acad. Sect. IA 49 (1973), 6-9 · Zbl 0268.32012 · doi:10.3792/pja/1195519485 [10] T. ODA, Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties, Ergebnisse der Math. (3) 15, Springer-Verlag, Berlin-New York, 1988. · Zbl 0628.52002 [11] M. SCHLESSINGER, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208-222 JSTOR: · Zbl 0167.49503 · doi:10.2307/1994967 [12] J. H. M. STEENBRINK, Logarithmic embeddings of varieties with normal crossings and mixed Hodg structures, Math. Ann. 301 (1995), 105-118. · Zbl 0814.14010 · doi:10.1007/BF01446621 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.