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Log structure : ウィキペディア英語版
Log structure

In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has applications in the theory of moduli spaces, in deformation theory and Fontaine's p-adic Hodge theory, among others.
== Motivation ==
The idea is to study some algebraic variety (or scheme) ''U'' which is smooth but not necessarily proper by embedding it into ''X'', which is proper, and then looking at certain sheaves on ''X''. The problem is that the subsheaf of \mathcal_X consisting of functions whose restriction to ''U'' is invertible is not a sheaf of rings (as adding two non-vanishing functions could provide one which vanishes), and we only get a sheaf of submonoids of \mathcal_X , multiplicatively. Remembering this additional structure on ''X'' corresponds to somehow remembering the inclusion j \colon U \to X , which likens ''X'' with this extra structure to a variety with boundary (corresponding to D = X - U ).〔Arthur Ogus (2011). Lectures on Logarithmic Algebraic Geometry.〕

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