Logarithmically regular morphisms
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Mathematische Annalen
Logarithmically regular morphisms Sam Molcho1 · Michael Temkin1 Received: 13 February 2020 / Revised: 14 October 2020 / Accepted: 4 November 2020 © The Author(s) 2020
Abstract We consider the stack Log X parametrizing log schemes over a log scheme X , and weak and strong properties of log morphisms via Log X , as defined by Olsson. We give a concrete combinatorial presentation of Log X , and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato’s chart criterion of logarithmic smoothness.
1 Introduction 1.1 Olsson’s approach to log geometry In [10], Olsson defined and studied the stack Log X of logarithmic schemes over a given fine logarithmic scheme X : Objects of Log X are fine log schemes over X and morphisms between two fine log schemes are morphisms over X that are strict, that is, which preserve the log structure. One of Olsson’s key insights was to apply the construction of Log X to the study of morphisms of logarithmic schemes. According to Olsson, a morphism f : X → Y of fine log schemes should have a “log” property P if the associated morphism Log f : Log X → LogY has the same property P. There is however a second natural possible definition, in which one requests that
Communicated by Jean-Yves Welschinger. This research is supported by ERC Consolidator Grant 770922-BirNonArchGeom. The project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (Grant agreement no. 786580). We would like to thank the anonymous referee for valuable comments.
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Michael Temkin [email protected] Sam Molcho [email protected]
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Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv’at Ram, 91904 Jerusalem, Israel
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S. Molcho, M. Temkin
the composed morphism X → Log X → LogY has property P. Morphisms f that satisfy this second definition are said to satisfy “weak log” property P. In general, the two notions do not coincide, and each has its distinct set of advantages. The strong property has better functorial behavior, but is much harder to work with explicitly. Olsson showed in [10] that the strong and weak properties coincide for three important classes of morphisms, namely log étale, log smooth, and log flat morphisms. All three classes had been introduced before by Kato, the first two in [7], and log flatness in an unpublished article. The definitions differ in nature however. Log étaleness and log smoothness were defined intrinsically, via a logarithmic analogue of the classical infinitesimal lifting property that defines étale and smooth morphisms. Then, Kato proved an extremely useful “chart” criterion which characterizes the local behavior of log smooth and étale maps. On the other hand, logarithmic flatness did not possess an intrinsic definition, and was defined by using a modification of the chart
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