# Log structure > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Log_structure > Markdown URL: https://mediated.wiki/source/Log_structure.md > Source: https://en.wikipedia.org/wiki/Log_structure > Source revision: 1356361179 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{for|log buildings|log building}} In [algebraic geometry](/source/algebraic_geometry), a '''log structure''' provides an abstract context to study [semistable scheme](/source/semistable_scheme)s, and in particular the notion of [logarithmic differential form](/source/logarithmic_form) and the related [Hodge-theoretic](/source/Hodge_theory) concepts. This idea is one of the foundations of '''logarithmic geometry''' and has applications in the theory of [moduli spaces](/source/moduli_spaces), in [deformation theory](/source/deformation_theory) and Fontaine's [p-adic Hodge theory](/source/p-adic_Hodge_theory), among others. == Motivation == The idea is to study some [algebraic variety](/source/algebraic_variety) (or [scheme](/source/scheme_(mathematics))) ''U'' which is [smooth](/source/smooth_morphism) but not necessarily [proper](/source/proper_morphism) by embedding it into ''X'', which is proper, and then looking at certain sheaves on ''X''. The problem is that the subsheaf of \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 sub[monoid](/source/monoid)s of \mathcal{O}_X , multiplicatively. Remembering this additional structure on ''X'' corresponds to 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](/source/Arthur_Ogus) (2011). Lectures on Logarithmic Algebraic Geometry. == Definition == Let ''X'' be a scheme. A '''pre-log structure''' on ''X'' consists of a sheaf of (commutative) monoids \mathcal{M} on ''X'' together with a homomorphism of monoids \alpha \colon \mathcal{M} \to \mathcal{O}_X , where \mathcal{O}_X is considered as a monoid under multiplication of functions. A pre-log structure (\mathcal{M}, \alpha) is a '''log structure''' if in addition \alpha induces an isomorphism \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 \mathcal{O}_X. A log scheme is simply a scheme furnished with a log structure. == Examples == *For any scheme ''X'', one can define the ''trivial log structure'' on ''X'' by taking \mathcal{M} = \mathcal{O}_X^\times and \alpha to be the inclusion. *The motivating example for the definition of log structure comes from semistable schemes. Let ''X'' be a scheme, j \colon U \to X the inclusion of an open subscheme of ''X'', with complement D = X - U a [divisor with normal crossings](/source/Normal_crossings). Then there is a log structure associated to this situation, which is \mathcal{M} = \mathcal{O}_X \cap j_* \mathcal{O}_U^\times, with \alpha simply the inclusion morphism into \mathcal{O}_X. This is called the ''canonical'' (or ''standard'') ''log structure'' on ''X'' associated to ''D''. * Let ''R'' be a [discrete valuation ring](/source/discrete_valuation_ring), with residue field ''k'' and fraction field ''K''. Then the ''canonical log structure'' on \mathrm{Spec}(R) consists of the inclusion of R \setminus \{ 0 \} (and not R^\times !) inside R. This is in fact an instance of the previous construction, but taking j \colon \mathrm{Spec}(K) \to \mathrm{Spec}(R). * With ''R'' as above, one can also define the ''hollow log structure'' on \mathrm{Spec}(R) by taking the same sheaf of monoids as previously, but instead sending the maximal ideal of ''R'' to 0. == Applications == One application of log structures is the ability to define [logarithmic form](/source/logarithmic_form)s (also called differential forms with log poles) on any log scheme. From this, one can for instance define log-smoothness and log-étaleness, generalizing the notions of [smooth morphism](/source/smooth_morphism)s and [étale morphism](/source/%C3%A9tale_morphism)s. This then allows the study of [deformation theory](/source/deformation_theory). In addition, log structures serve to define the [mixed Hodge structure](/source/mixed_Hodge_structure) on any smooth complex variety ''X'', by taking a compactification with boundary a normal crossings divisor ''D'', and writing down the corresponding [logarithmic de Rham complex](/source/logarithmic_form).Chris A.M. Peters; Joseph H.M. Steenbrink (2008). Mixed Hodge Structures. Springer. {{isbn|978-3-540-77015-2}} Log objects also naturally appear as the objects at the boundary of [moduli space](/source/moduli_space)s, i.e. from degenerations. Log geometry also allows the definition of [log-crystalline cohomology](/source/log-crystalline_cohomology), an analogue of [crystalline cohomology](/source/crystalline_cohomology) which has good behaviour for varieties that are not necessarily smooth, only log smooth. This then has application to the theory of [Galois representation](/source/Galois_representation)s, and particularly semistable Galois representations. == See also == *[Log geometry](/source/Log_geometry) *[Semistable scheme](/source/Semistable_scheme) *[Log-crystalline cohomology](/source/Log-crystalline_cohomology) == References == {{Reflist}} Category:Algebraic geometry Category:Scheme theory --- Adapted from the Wikipedia article [Log structure](https://en.wikipedia.org/wiki/Log_structure) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Log_structure?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.