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 is one of the foundations of logarithmic geometry and 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 O X {\displaystyle {\mathcal {O}}_{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 O X {\displaystyle {\mathcal {O}}_{X}} , multiplicatively. Remembering this additional structure on X corresponds to remembering the inclusion j : U → X {\displaystyle j\colon U\to X} , which likens X with this extra structure to a variety with boundary (corresponding to D = X − U {\displaystyle D=X-U} ).
Definition Let X be a scheme. A pre-log structure on X consists of a sheaf of (commutative) monoids M {\displaystyle {\mathcal {M}}} on X together with a homomorphism of monoids α : M → O X {\displaystyle \alpha \colon {\mathcal {M}}\to {\mathcal {O}}_{X}} , where O X {\displaystyle {\mathcal {O}}_{X}} is considered as a monoid under multiplication of functions. A pre-log structure ( M , α ) {\displaystyle ({\mathcal {M}},\alpha )} is a log structure if in addition α {\displaystyle \alpha } induces an isomorphism α : α − 1 ( O X × ) → O X × {\displaystyle \alpha \colon \alpha ^{-1}({\mathcal {O}}_{X}^{\times })\to {\mathcal {O}}_{X}^{\times }} . A morphism of (pre-)log structures consists in a homomorphism of sheaves of monoids commuting with the associated homomorphisms into O X {\displaystyle {\mathcal {O}}_{X}} . A log scheme is simply a scheme furnished with a log structure.
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