## Log Smooth Deformation Theory (1994)

Venue: | Tohoku Math. J |

Citations: | 25 - 4 self |

### BibTeX

@ARTICLE{Kato94logsmooth,

author = {Fumiharu Kato},

title = {Log Smooth Deformation Theory},

journal = {Tohoku Math. J},

year = {1994},

volume = {48},

pages = {317--354}

}

### OpenURL

### Abstract

this article, we formulate and develop the theory of log smooth deformations. Here, log smoothness (more precisely, logarithmic smoothness) is a concept in log geometry

### Citations

197 | Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 15 - Oda - 1988 |

155 |
Functors of Artin Rings
- Schlessinger
- 1968
(Show Context)
Citation Context ... construct log smooth deformation functor by the concept of infinitesimal log smooth lifting. The goal of this paper is to show that this functor has a representable hull in the sense of Schlessinger =-=[11]-=-, under certain conditions (Theorem 8.7). At the end of this paper, we give two examples of our log smooth deformation theory, which are summarized as follows: 1. Deformations with divisors (x10): Let... |

140 |
Logarithmic Structures of Fontaine-Illusie
- Kato
- 1990
(Show Context)
Citation Context ... unifies the scheme theory and the theory of toric varieties. This theory was initiated by Fontaine and Illusie, based on their idea of log structures on schemes, and further developed by Kazuya Kato =-=[5]-=-. Recently, the importance of log geometry has come to be recognized by many geometers and applied to various fields of algebraic and arithmetic geometry. One of such applications can be seen in the r... |

134 |
Revetements etales et groupe fondamental
- Grothendieck
- 1971
(Show Context)
Citation Context ...e X;O e X )\Omega A I, D induces an automorphism of O e X 0 and Dlog induces an automorphism of f M 0 , and then, indues an automorphism of ( e X 0 ; f M 0 ). By this, applying the arguement in SGA I =-=[2]-=- Expos'e 3, we get the following proposition. Proposition 8.4 (cf. [5, (3.14)]) Let e f : ( e X; f M) ! (Spec A; Q) be a log smooth lifting of f to A, and u : A 0 ! A a surjective homomorphism in C 3[... |

58 | On deformations of complex analytic structures I - Kodaira, Morrow, et al. - 1971 |

48 |
Global smoothings of varieties with normal crossings
- Friedman
- 1983
(Show Context)
Citation Context ...er a field k is, 'etale locally, isomorphic to an affine normal crossing variety Spec k[z 1 ; . . . ; zn ]=(z 1 1 1 1 z l ), then we call X a normal crossing variety over k. If X is d-semistable (cf. =-=[1]-=-), then there exists a log structure M on X of semistable type (Definition 11.6) and (X; M) is log smooth over a standard log point (Spec k; N) (Theorem 11.7). Then, a log smooth deformation in our se... |

31 |
Logarithmic deformations of normal crossing varieties and smoothings of degenerate Calabi-Yau varieties
- KAWAMATA, NAMIKAWA
(Show Context)
Citation Context ...5 For any logarithmic embedding (X; M), there exists an exact sequence of abelian sheaves 1 0! O 2 X 0! M gp 0!s3 Z e X 0! 0; (11) where : e X ! X is the normalization of X. Definition 11.6 (cf. [4], =-=[6]-=-) A log structure of embedding type M! OX is said to be of semistable type, if there exists a homomorphism ZX !M gp of abelian sheaves on X such that the diagram M gp 0!s3 Z e X - % d ZX commutes, whe... |

25 |
The cotangent complex of a morphism
- Lichtenbaum, Schlessiger
(Show Context)
Citation Context ...ae A. The cotangent complex of the morphism k ! A is given by L ffl : 0 0! R\Omega R A ffi 0!\Omega 1 R=k\Omega R A 0! 0; where ffi is defined by R ! F 1 R d !\Omega 1 R=k with F = Z 1 1 1 1 Z l (cf. =-=[8]-=-). Then the tangent complex of U is the complex HomA (L ffl ; A): 0 0! 2 R=k\Omega R A ffi 3 0! HomA (R\Omega R A; A) 0! 0; where 2 R=k = HomR(\Omega 1 R=k ; R). We define T 1 A = HomA (R\Omega R A; A... |

5 | Revêtements étales et groupe fondamentale - Grothendieck - 1971 |

5 |
Logarithmic compactifications of the generalized jacobian variety
- Kajiwara
- 1993
(Show Context)
Citation Context ...n 11.5 For any logarithmic embedding (X; M), there exists an exact sequence of abelian sheaves 1 0! O 2 X 0! M gp 0!s3 Z e X 0! 0; (11) where : e X ! X is the normalization of X. Definition 11.6 (cf. =-=[4]-=-, [6]) A log structure of embedding type M! OX is said to be of semistable type, if there exists a homomorphism ZX !M gp of abelian sheaves on X such that the diagram M gp 0!s3 Z e X - % d ZX commutes... |

5 |
Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures
- Steenbrink
- 1995
(Show Context)
Citation Context ...e of log geometry has come to be recognized by many geometers and applied to various fields of algebraic and arithmetic geometry. One of such applications can be seen in the recent work of Steenbrink =-=[12]-=-. In the present paper, we attempt to apply log geometry to extend the usual smooth deformation theory by using the concept of log smoothness. Log smoothness is one of the most important concepts in l... |

2 |
Introduction à la géométrie logarithmique, Seminar notes at Tokyo Univ
- Illusie
- 1992
(Show Context)
Citation Context ...review the definition and basic properties of log smoothness in Section 3. In Section 4, we study the characterization of log smoothness by means of the theory of toric varieties according to Illusie =-=[3]-=- and Kato [5]. In Section 5, we recall the definitions and basic properties of log derivations and log differentials. In Sections 6 and 7, we give the proofs of the theorems stated in Section 4. Secti... |

1 |
On the relative pseudo--rigidity
- Makio
- 1973
(Show Context)
Citation Context ...mooth deformation in our sense is a deformation of 1 the pair (X; D). If X itself is smooth and D is a smooth divisor on X , our deformations coincides with the relative deformations studied by Makio =-=[9]-=- and others. 2. Smoothings of normal crossing varieties (x11): If a scheme of finite type X over a field k is, 'etale locally, isomorphic to an affine normal crossing variety Spec k[z 1 ; . . . ; zn ]... |