# zbMATH — the first resource for mathematics

Logarithmic geometry and algebraic stacks. (English) Zbl 1069.14022
This paper deals with logarithmic geometry (or logarithmic spaces) in the sense of K. Kato [in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191–224 (1989; Zbl 0776.14004)]. Throughout the paper, in which the author assumes that the reader is familiar with the basics of this conceptual framework [L. Illusie, in: Barsotti symposium in algebraic geometry (Abano Terme, 1991), Perspect. Math. 15, 183–204 (1994; Zbl 0832.14015)], a log structure on a scheme $$X$$ means a log structure on the étale site $$X_{\text{ét}}$$ of $$X$$ in the sense of Kato.
In this context, the purpose of the present paper is to introduce a stack-theoretic approach to Kato’s theory of logarithmic structures. More precisely, for any fine log scheme $$S$$ with underlying scheme $$\mathring S$$, the author constructs a fibred category $$\text{Log}_S\to (\mathring S$$-schemes). His main result consists in the proof of the fundamental theorem stating that $$\text{Log}_S$$ is an algebraic stack locally of finite presentation over the underlying scheme $$\mathring S$$. Moreover, it is shown that a morphism of fine log schemes $$f:X\to S$$ defines tautologically a morphism $$\text{Log}(f) : \text{Log}_X\to \text{Log}_S$$ of algebraic stacks, and that the association $$S\mapsto \text{Log}_S$$ defines a 2-functor from the category of log schemes to the 2-category of algebraic stacks. It is then explained how this 2-functor can be used to reinterpret and study some original basic notions in Kato’s logarithmic geometry.
The fine analysis carried out in this paper is enhanced by an appendix, in which the author compares the notions of log structure in the fppf, étale, and Zariski topology, respectively. Part of this comparison is used in the course of the main body of the paper, and the rest is included for the sake of completeness.
As the author points out, his main theorem on the structure of the stack $$\text{Log}_S$$ has further applications which are not discussed in the present paper. Namely, one can develop the theory of log crystalline cohomology using a theory of crystalline cohomology over stacks [cf. M. C. Olsson, Crystalline cohomology of schemes over algebraic stacks, Preprint 2002], and also the deformation theory of log schemes can be analyzed using the structure of $$\text{Log}_S$$. In addition, the main theorem of the present paper has a natural place in the moduli theory of fine log schemes [cf. M. C. Olsson, Tohoku Math. J., II. Ser. 55, No. 3, 397–438 (2003; Zbl 1069.14015)]. The author intends to discuss these subjects more thoroughly in forthcoming papers.

##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14A20 Generalizations (algebraic spaces, stacks) 14A15 Schemes and morphisms 14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc. 18D30 Fibered categories
Full Text:
##### References:
 [1] Artin M. , Versal deformations and algebraic stacks , Invent. Math. 27 ( 1974 ) 165 - 189 . MR 399094 | Zbl 0317.14001 · Zbl 0317.14001 [2] Artin M. , Bertin J.E. , Demazure M. , Gabriel P. , Grothendieck A. , Raynaud M. , Serre J.-P. , Schémas en groupes. Fasc. 1: Exposés 1 à 4 , Institut des Hautes Études Scientifiques , Paris , 1963/1964 . MR 207702 [3] Artin M. , Grothendieck A. , Verdier J.-L. , Théorie des topos et cohomologie étale des schémas , in: Artin M. , Grothendieck A. , Verdier J.L. (Eds.), Séminaire de géométrie algébrique du Bois-Marie 1963-1964 (SGA 4) , Lecture Notes in Mathematics , Vols. 269, 270, 305 , Springer-Verlag , Berlin , 1972 , Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 354654 | Zbl 0237.00012 · Zbl 0237.00012 [4] Dieudonné J. , Grothendieck A. , Éléments de géométrie algébrique, no. 4, 8, 11, 17, 20, 24, 28, 32 , Inst. Hautes Études Sci. Publ. Math , Paris , 1961 . Numdam [5] Grothendieck A. , Revêtements étale et groupe fondamental , Lecture Notes in Mathematics , vol. 224 , Springer-Verlag , Berlin , 1971 . MR 354651 [6] Hartshorne R. , Algebraic Geometry , Graduate Texts in Mathematics , vol. 52 , Springer-Verlag , New York , 1977 . MR 463157 | Zbl 0367.14001 · Zbl 0367.14001 [7] Illusie L. , Logarithmic spaces (according to K. Kato) , in: Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) , Academic Press , San Diego, CA , 1994 , pp. 183 - 203 . MR 1307397 | Zbl 0832.14015 · Zbl 0832.14015 [8] Kato F. , Log smooth deformation theory , Tôhoku Math. J. (2) 48 ( 3 ) ( 1996 ) 317 - 354 . Article | MR 1404507 | Zbl 0876.14007 · Zbl 0876.14007 [9] Kato K. , Logarithmic structures of Fontaine-Illusie , in: Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988) , Johns Hopkins Univ. Press , Baltimore, MD , 1989 , pp. 191 - 224 . MR 1463703 | Zbl 0776.14004 · Zbl 0776.14004 [10] Kato K., Logarithmic structures of Fontaine-Illusie II: Logarithmic flat topology (incomplete) preprint, 1991. MR 1463703 [11] Kato K., Log flat morphisms and neat charts, manuscript notes, 1991. [12] Knutson D. , Algebraic Spaces , Lecture Notes in Mathematics , vol. 203 , Springer-Verlag , Berlin , 1971 . MR 302647 | Zbl 0221.14001 · Zbl 0221.14001 [13] Lafforgue L. , Chtoucas de Drinfeld et correspondance de Langlands , Invent. Math. 147 ( 2002 ) 1 - 241 . MR 1875184 | Zbl 1038.11075 · Zbl 1038.11075 [14] Laumon G. , Moret-Bailly L. , Champs algébriques , Ergeb. Math. 39 ( 2000 ). MR 1771927 · Zbl 0945.14005 [15] Milne J.S. , Étale Cohomology , Princeton University Press , Princeton, NJ , 1980 . MR 559531 | Zbl 0433.14012 · Zbl 0433.14012 [16] Niziol W., Toric singularities: log-blow-ups and global resolutions , preprint, 1999. MR 2177194 [17] Ogus A., Lectures on logarithmic algebraic geometry , TeXed notes, 2001. [18] Olsson M., Log algebraic stacks and moduli of log schemes , PhD Thesis, UC Berkeley, May 2001. [19] Olsson M., Crystalline cohomology of schemes over algebraic stacks , preprint, 2002. [20] Olsson M. , Universal log structures on semi-stable varieties , Tôhoku Math. J. 55 ( 2003 ) 397 - 438 . Article | MR 1993863 | Zbl 1069.14015 · Zbl 1069.14015
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.