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<title>Log structure</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Log structure</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For log buildings, see <a href="Log_building" title="Log building">log building</a>.</div>

<p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <b>log structure</b> provides an abstract context to study semistable schemes, and in particular the notion of <a href="Logarithmic_form" title="Logarithmic form">logarithmic differential form</a> and the related <a href="Hodge_theory" title="Hodge theory">Hodge-theoretic</a> concepts. This idea has applications in the theory of <a href="Moduli_spaces" class="mw-redirect" title="Moduli spaces">moduli spaces</a>, in <a href="Deformation_theory" class="mw-redirect" title="Deformation theory">deformation theory</a> and Fontaine's <a href="P-adic_Hodge_theory" title="P-adic Hodge theory">p-adic Hodge theory</a>, among others.

</p>

<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

<p>The idea is to study some <a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a> (or <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>) <i>U</i> which is <a href="Smooth_morphism" title="Smooth morphism">smooth</a> but not necessarily <a href="Proper_morphism" title="Proper morphism">proper</a> by embedding it into <i>X</i>, which is proper, and then looking at certain sheaves on <i>X</i>. The problem is that the subsheaf of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

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</math></span><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span> consisting of functions whose restriction to <i>U</i> 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<a href="Monoid" title="Monoid">monoids</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

</semantics>

</math></span><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span>, multiplicatively. Remembering this additional structure on <i>X</i> corresponds to remembering the inclusion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\colon U\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\colon U\to X}</annotation>

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</math></span><img src="./3019dbb96536bfce42d383159aa5075b6399a00a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.395ex; height:2.509ex;" alt="{\displaystyle j\colon U\to X}" loading="lazy"></span>, which likens <i>X</i> with this extra structure to a variety with boundary (corresponding to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=X-U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=X-U}</annotation>

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</math></span><img src="./f4697d9cc82e6f599a01f6ffb19a162494cf6760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.626ex; height:2.343ex;" alt="{\displaystyle D=X-U}" loading="lazy"></span>).<sup id="cite_ref-Ogus_1-0" class="reference"><a href="#cite_note-Ogus-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>Let <i>X</i> be a scheme. A <b>pre-log structure</b> on <i>X</i> consists of a sheaf of (commutative) monoids <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}}</annotation>

</semantics>

</math></span><img src="./2cc2abebd45ec020509a0ec548b67c9a2cb7cecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle {\mathcal {M}}}" loading="lazy"></span> on <i>X</i> together with a homomorphism of monoids <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon {\mathcal {M}}\to {\mathcal {O}}_{X}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \colon {\mathcal {M}}\to {\mathcal {O}}_{X}}</annotation>

</semantics>

</math></span><img src="./3b12cd69271195909853a3f7233cb69eaa7b3533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.409ex; height:2.509ex;" alt="{\displaystyle \alpha \colon {\mathcal {M}}\to {\mathcal {O}}_{X}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

</semantics>

</math></span><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span> is considered as a monoid under multiplication of functions.

</p><p>A pre-log structure <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathcal {M}},\alpha )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\mathcal {M}},\alpha )}</annotation>

</semantics>

</math></span><img src="./3baff8ef7e53ad6c705f5f74256b9d321d475f0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.121ex; height:2.843ex;" alt="{\displaystyle ({\mathcal {M}},\alpha )}" loading="lazy"></span> is a <b>log structure</b> if in addition <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> induces an isomorphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon \alpha ^{-1}({\mathcal {O}}_{X}^{\times })\to {\mathcal {O}}_{X}^{\times }}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \colon \alpha ^{-1}({\mathcal {O}}_{X}^{\times })\to {\mathcal {O}}_{X}^{\times }}</annotation>

</semantics>

</math></span><img src="./a5f074c8c6c5dcc9180b0499a9d24b8af35d832a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.73ex; height:3.176ex;" alt="{\displaystyle \alpha \colon \alpha ^{-1}({\mathcal {O}}_{X}^{\times })\to {\mathcal {O}}_{X}^{\times }}" loading="lazy"></span>.

</p><p>A morphism of (pre-)log structures consists in a homomorphism of sheaves of monoids commuting with the associated homomorphisms into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

</semantics>

</math></span><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span>.

</p><p>A log scheme is simply a scheme furnished with a log structure.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>For any scheme <i>X</i>, one can define the <i>trivial log structure</i> on <i>X</i> by taking <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}={\mathcal {O}}_{X}^{\times }}">

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}={\mathcal {O}}_{X}^{\times }}</annotation>

</semantics>

</math></span><img src="./39944d7ef7e7a23b2d4ee94a3de87e44f3880b4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.371ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}={\mathcal {O}}_{X}^{\times }}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> to be the inclusion.</li>

<li>The motivating example for the definition of log structure comes from semistable schemes. Let <i>X</i> be a scheme, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\colon U\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\colon U\to X}</annotation>

</semantics>

</math></span><img src="./3019dbb96536bfce42d383159aa5075b6399a00a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.395ex; height:2.509ex;" alt="{\displaystyle j\colon U\to X}" loading="lazy"></span> the inclusion of an open subscheme of <i>X</i>, with complement <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=X-U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=X-U}</annotation>

</semantics>

</math></span><img src="./f4697d9cc82e6f599a01f6ffb19a162494cf6760.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.626ex; height:2.343ex;" alt="{\displaystyle D=X-U}" loading="lazy"></span> a <a href="Normal_crossings" class="mw-redirect" title="Normal crossings">divisor with normal crossings</a>. Then there is a log structure associated to this situation, which is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}={\mathcal {O}}_{X}\cap j_{*}{\mathcal {O}}_{U}^{\times }}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}={\mathcal {O}}_{X}\cap j_{*}{\mathcal {O}}_{U}^{\times }}</annotation>

</semantics>

</math></span><img src="./b0b9a641f8dbd6cc8fc0df1968eab57fffce7539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.327ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}={\mathcal {O}}_{X}\cap j_{*}{\mathcal {O}}_{U}^{\times }}" loading="lazy"></span>, with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> simply the inclusion morphism into <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

</semantics>

</math></span><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"></span>. This is called the <i>canonical</i> (or <i>standard</i>) <i>log structure</i> on <i>X</i> associated to <i>D</i>.</li>

<li>Let <i>R</i> be a <a href="Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a>, with residue field <i>k</i> and fraction field <i>K</i>. Then the <i>canonical log structure</i> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Spec} (R)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Spec} (R)}</annotation>

</semantics>

</math></span><img src="./b3eb3e8baab788aae477cd5fb3f4adde00599134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.223ex; height:2.843ex;" alt="{\displaystyle \mathrm {Spec} (R)}" loading="lazy"></span> consists of the inclusion of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\setminus \{0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\setminus \{0\}}</annotation>

</semantics>

</math></span><img src="./cadb5bd1742f04208ee7dbbafa2f040b32dc2d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.446ex; height:2.843ex;" alt="{\displaystyle R\setminus \{0\}}" loading="lazy"></span> (and not <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\times }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{\times }}</annotation>

</semantics>

</math></span><img src="./92c9000bc6f7ffcd09966ca0187b63c877c0d5e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.275ex; height:2.343ex;" alt="{\displaystyle R^{\times }}" loading="lazy"></span>!) inside <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>. This is in fact an instance of the previous construction, but taking <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\colon \mathrm {Spec} (K)\to \mathrm {Spec} (R)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\colon \mathrm {Spec} (K)\to \mathrm {Spec} (R)}</annotation>

</semantics>

</math></span><img src="./732dd47e4f15fa371c24a9b5413b1e8f27c2e1e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:22.381ex; height:2.843ex;" alt="{\displaystyle j\colon \mathrm {Spec} (K)\to \mathrm {Spec} (R)}" loading="lazy"></span>.</li>

<li>With <i>R</i> as above, one can also define the <i>hollow log structure</i> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Spec} (R)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Spec} (R)}</annotation>

</semantics>

</math></span><img src="./b3eb3e8baab788aae477cd5fb3f4adde00599134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.223ex; height:2.843ex;" alt="{\displaystyle \mathrm {Spec} (R)}" loading="lazy"></span> by taking the same sheaf of monoids as previously, but instead sending the maximal ideal of <i>R</i> to 0.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>One application of log structures is the ability to define <a href="Logarithmic_form" title="Logarithmic form">logarithmic forms</a> (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 <a href="Smooth_morphism" title="Smooth morphism">smooth morphisms</a> and <a href="%C3%89tale_morphism" title="Étale morphism">étale morphisms</a>. This then allows the study of <a href="Deformation_theory" class="mw-redirect" title="Deformation theory">deformation theory</a>.

</p><p>In addition, log structures serve to define the <a href="Mixed_Hodge_structure" title="Mixed Hodge structure">mixed Hodge structure</a> on any smooth complex variety <i>X</i>, by taking a compactification with boundary a normal crossings divisor <i>D</i>, and writing down the corresponding <a href="Logarithmic_form" title="Logarithmic form">logarithmic de Rham complex</a>.<sup id="cite_ref-MHS_2-0" class="reference"><a href="#cite_note-MHS-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Log objects also naturally appear as the objects at the boundary of <a href="Moduli_space" title="Moduli space">moduli spaces</a>, i.e. from degenerations.

</p><p>Log geometry also allows the definition of log-crystalline cohomology, an analogue of <a href="Crystalline_cohomology" title="Crystalline cohomology">crystalline cohomology</a> which has good behaviour for varieties that are not necessarily smooth, only log smooth. This then has application to the theory of <a href="Galois_representation" title="Galois representation">Galois representations</a>, and particularly semistable Galois representations.

</p>

<ul><li>Log geometry</li>

<li>Semistable scheme</li>

<li>Log-crystalline cohomology</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-Ogus-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ogus_1-0">^</a></b></span> <span class="reference-text"><a href="Arthur_Ogus" title="Arthur Ogus">Arthur Ogus</a> (2011). Lectures on Logarithmic Algebraic Geometry.</span>

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<li id="cite_note-MHS-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-MHS_2-0">^</a></b></span> <span class="reference-text">Chris A.M. Peters; Joseph H.M. Steenbrink (2008). Mixed Hodge Structures. Springer. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-77015-2</bdi></span>

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This article is issued from <a class="external text" title="Last edited on 2025-07-24" href="https://en.wikipedia.org/wiki/?title=Log_structure&oldid=1302311926">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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