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In `F33f`_`[algebraic geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_geometry]`_`f, a `!log structure`! provides an abstract context to study semistable schemes, and in particular the notion of `F33f`_`[logarithmic differential form`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmic_form]`_`f and the related `F33f`_`[Hodge-theoretic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hodge_theory]`_`f concepts. This idea has applications in the theory of `F33f`_`[moduli spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Moduli_spaces]`_`f, in `F33f`_`[deformation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Deformation_theory]`_`f and Fontaine's `F33f`_`[p-adic Hodge theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=P-adic_Hodge_theory]`_`f, among others.

>>Contents

• `F0af`_`[Motivation`#motivation]`_`f

• `F0af`_`[Definition`#definition]`_`f

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Applications`#applications]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Motivation

The idea is to study some `F33f`_`[algebraic variety`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_variety]`_`f (or `F33f`_`[scheme`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Scheme_(mathematics)]`_`f) `*U`* which is `F33f`_`[smooth`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_morphism]`_`f but not necessarily `F33f`_`[proper`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Proper_morphism]`_`f 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 sub`F33f`_`[monoids`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monoid]`_`f 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} ).`:cite-ref-ogus-1-0[`F5bf`_`[1`#cite-note-ogus-1]`_`f]

>>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 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 α ) {\\displaystyle ({\\mathcal {M}},\\alpha )} is a `!log structure`! if in addition α α {\\displaystyle \\alpha } induces an isomorphism α α : : 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.

>>Examples

• For any scheme `*X`*, one can define the `*trivial log structure`* on `*X`* by taking M = O X × × {\\displaystyle {\\mathcal {M}}={\\mathcal {O}}_{X}^{\\times }} and α α {\\displaystyle \\alpha } to be the inclusion.

• The motivating example for the definition of log structure comes from semistable schemes. Let `*X`* be a scheme, j : : U → → X {\\displaystyle j\\colon U\\to X} the inclusion of an open subscheme of `*X`*, with complement D = X − − U {\\displaystyle D=X-U} a `F33f`_`[divisor with normal crossings`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_crossings]`_`f. Then there is a log structure associated to this situation, which is M = O X ∩ ∩ j ∗ ∗ O U × × {\\displaystyle {\\mathcal {M}}={\\mathcal {O}}_{X}\\cap j_{*}{\\mathcal {O}}_{U}^{\\times }} , with α α {\\displaystyle \\alpha } simply the inclusion morphism into O X {\\displaystyle {\\mathcal {O}}_{X}} . This is called the `*canonical`* (or `*standard`*) `*log structure`* on `*X`* associated to `*D`*.

• Let `*R`* be a `F33f`_`[discrete valuation ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Discrete_valuation_ring]`_`f, with residue field `*k`* and fraction field `*K`*. Then the `*canonical log structure`* on S p e c ( R ) {\\displaystyle \\mathrm {Spec} (R)} consists of the inclusion of R ∖ ∖ { 0 } {\\displaystyle R\\setminus \\{0\\}} (and not R × × {\\displaystyle R^{\\times }} !) inside R {\\displaystyle R} . This is in fact an instance of the previous construction, but taking j : : S p e c ( K ) → → S p e c ( R ) {\\displaystyle j\\colon \\mathrm {Spec} (K)\\to \\mathrm {Spec} (R)} .

• With `*R`* as above, one can also define the `*hollow log structure`* on S p e c ( R ) {\\displaystyle \\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 `F33f`_`[logarithmic forms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmic_form]`_`f (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 `F33f`_`[smooth morphisms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Smooth_morphism]`_`f and `F33f`_`[étale morphisms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Étale_morphism]`_`f. This then allows the study of `F33f`_`[deformation theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Deformation_theory]`_`f.

In addition, log structures serve to define the `F33f`_`[mixed Hodge structure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mixed_Hodge_structure]`_`f on any smooth complex variety `*X`*, by taking a compactification with boundary a normal crossings divisor `*D`*, and writing down the corresponding `F33f`_`[logarithmic de Rham complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmic_form]`_`f.`:cite-ref-mhs-2-0[`F5bf`_`[2`#cite-note-mhs-2]`_`f]

Log objects also naturally appear as the objects at the boundary of `F33f`_`[moduli spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Moduli_space]`_`f, i.e. from degenerations.

Log geometry also allows the definition of log-crystalline cohomology, an analogue of `F33f`_`[crystalline cohomology`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Crystalline_cohomology]`_`f which has good behaviour for varieties that are not necessarily smooth, only log smooth. This then has application to the theory of `F33f`_`[Galois representations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Galois_representation]`_`f, and particularly semistable Galois representations.

>>See also

• Log geometry

• Semistable scheme

• Log-crystalline cohomology

>>References

`:cite-note-ogus-1`!1.`! `F0af`_`[↑`#cite-ref-ogus-1-0]`_`f `F33f`_`[Arthur Ogus`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arthur_Ogus]`_`f (2011). Lectures on Logarithmic Algebraic Geometry.

`:cite-note-mhs-2`!2.`! `F0af`_`[↑`#cite-ref-mhs-2-0]`_`f Chris A.M. Peters; Joseph H.M. Steenbrink (2008). Mixed Hodge Structures. Springer. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-77015-2

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