Log smooth deformation theory. (English) Zbl 0876.14007

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)


14D15 Formal methods and deformations in algebraic geometry
13D10 Deformations and infinitesimal methods in commutative ring theory
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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[1] R. FRIEDMAN, Global smoothings of varieties with normal crossings, Ann. of Math. (2) 118 (1983), 75-114. JSTOR: · Zbl 0569.14002
[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
[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
[8] S. LICHTENBAUM AND M. ScHLESSiNGER, The cotangent complex of a morphism, Trans. Amer. Math Soc. 128 (1967), 41-70. JSTOR: · Zbl 0156.27201
[9] K. MAKIO, On the relative pseudo-rigidity, Proc. Japan Acad. Sect. IA 49 (1973), 6-9 · Zbl 0268.32012
[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
[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
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