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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="complex_geometry">Complex geometry</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+line">complex line</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="connections_on_bundles">Connections on bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>, <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> </ul> <h2 id="higher_abelian_differential_cohomology">Higher abelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+orientation">differential orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+Thom+class">differential Thom class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characters">differential characters</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe with connection</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> <h2 id="higher_nonabelian_differential_cohomology">Higher nonabelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h2 id="application_to_gauge_theory">Application to gauge theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>/<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/differential+cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToComplexCohomology'>Relation to complex cohomology</a></li> <li><a href='#filtering_and_relation_to_hodge_filtation'>Filtering and Relation to Hodge filtation</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>On a <a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, the <em>holomorphic de Rham complex</em> is the restriction of the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> to <a class="existingWikiWord" href="/nlab/show/holomorphic+differential+forms">holomorphic differential forms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^p</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>Σ</mi> <mo>•</mo></msubsup><mo>≔</mo><mrow><mo>(</mo><msub><mi>𝒪</mi> <mi>Σ</mi></msub><mover><mo>⟶</mo><mo>∂</mo></mover><msubsup><mi>Ω</mi> <mi>Σ</mi> <mn>1</mn></msubsup><mover><mo>⟶</mo><mo>∂</mo></mover><msubsup><mi>Ω</mi> <mi>Σ</mi> <mn>2</mn></msubsup><mo>⟶</mo><mi>⋯</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet_\Sigma \coloneqq \left( \mathcal{O}_\Sigma \stackrel{\partial}{\longrightarrow} \Omega^1_\Sigma \stackrel{\partial}{\longrightarrow} \Omega^2_\Sigma \longrightarrow \cdots \right) \,. </annotation></semantics></math></div> <p>Notice here that since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^k</annotation></semantics></math> denotes the holomorphic forms, this is the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the <a class="existingWikiWord" href="/nlab/show/Dolbeault+operator">Dolbeault operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar \partial</annotation></semantics></math> and hence the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>=</mo><mo>∂</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\mathbf{d} = \partial + \bar \partial</annotation></semantics></math> restricts to the holomorphic component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo></mrow><annotation encoding="application/x-tex">\partial</annotation></semantics></math> as above.</p> <p>Often considered is a version of this with singularities (“logarithmic de Rham complex”) where one consideres instead meromorphic differential forms which are holomorphic in the bulk with logarithmic singularities towards compactification boundaries of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="RelationToComplexCohomology">Relation to complex cohomology</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> then the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> is <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/hypercohomology">hyper</a>-<a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> with coefficients in the holomorphic de Rham complex:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>Σ</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^k(\Sigma,\mathbb{C}) \simeq H^k(\Sigma, \Omega_\Sigma^\bullet) </annotation></semantics></math></div> <p>(e.g. <a href="#Voisin02">Voisin 02, theorem 8.1</a>).</p> <h3 id="filtering_and_relation_to_hodge_filtation">Filtering and Relation to Hodge filtation</h3> <p>The holomorphic de Rham complex is naturally <a class="existingWikiWord" href="/nlab/show/filtered+object">filtered</a> by degree with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>th filtering stage being</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>p</mi></msup><msubsup><mi>Ω</mi> <mi>X</mi> <mo>•</mo></msubsup><mo>≔</mo><mo stretchy="false">(</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mi>p</mi></msubsup><mover><mo>⟶</mo><mo>∂</mo></mover><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mo>∂</mo></mover><mi>⋯</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^p \Omega^\bullet_X \coloneqq (0 \to \cdots \to \Omega^p_X \stackrel{\partial}{\longrightarrow} \Omega^{p+1} \stackrel{\partial}{\longrightarrow} \cdots) \,. </annotation></semantics></math></div> <p>Notice that here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^p</annotation></semantics></math> is still regarded as sitting in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">-p</annotation></semantics></math>, one just replaces by 0 in the holomorphic de Rham complex the groups of differential forms of degree less than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> <p>With this, the <em><a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(X,\mathbb{C})</annotation></semantics></math> is defined to be the <a class="existingWikiWord" href="/nlab/show/filtration">filtration</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>th stage the <a class="existingWikiWord" href="/nlab/show/image">image</a> of the <a class="existingWikiWord" href="/nlab/show/hypercohomology">hyper</a>-<a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> with coefficients in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>th filtering stage of the holomorphic de Rham complex inside that with coefficients the full de Rham complex:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mi>p</mi></msup><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>im</mi><mrow><mo>(</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>F</mi> <mi>p</mi></msup><msubsup><mi>Ω</mi> <mi>X</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> F^p H^k(X,\mathbb{C}) \coloneqq im \left( H^k(X, F^p \Omega^\bullet_X) \to H^k(X, \Omega^\bullet_X) \right) </annotation></semantics></math></div> <p>(e.g. <a href="#Voisin02">Voisin 02, def. 8.2</a>).</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> happens to be a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> then the <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+spectral+sequence">Frölicher spectral sequence</a> shows that this coincides with the more traditional definition via <a class="existingWikiWord" href="/nlab/show/harmonic+differential+forms">harmonic differential forms</a> (e.g. <a href="#Voisin02">Voisin 02, remark 8.29</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+form+with+logarithmic+singularities">differential form with logarithmic singularities</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+cohomology">Hodge cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Voisin02"> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, section 8.2 of <em><a class="existingWikiWord" href="/nlab/show/Hodge+theory+and+Complex+algebraic+geometry">Hodge theory and Complex algebraic geometry</a> I,II</em>, Cambridge Stud. in Adv. Math. <strong>76, 77</strong>, 2002/3</p> </li> <li id="Beilinson85"> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Alexander Beilinson</a>, <em>Higher regulators and values of L-functions</em>, J. Soviet Math. 30 (1985), 2036-2070 (reviewed in <a href="#EsnaultViehweg88">Esnault-Viehweg 88</a>) (<a href="http://dx.doi.org/10.1007/BF02105861">doi</a>)</p> </li> <li id="EsnaultViehweg88"> <p><a class="existingWikiWord" href="/nlab/show/H%C3%A9l%C3%A8ne+Esnault">Hélène Esnault</a>, <a class="existingWikiWord" href="/nlab/show/Eckart+Viehweg">Eckart Viehweg</a>, section 2.5 of: <em>Deligne-Beilinson cohomology</em>, in: <a class="existingWikiWord" href="/nlab/show/Michael+Rapoport">Michael Rapoport</a>, <a class="existingWikiWord" href="/nlab/show/Norbert+Schappacher">Norbert Schappacher</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Schneider">Peter Schneider</a> (eds.), <em><a class="existingWikiWord" href="/nlab/show/Beilinson%27s+Conjectures+on+Special+Values+of+L-Functions">Beilinson's Conjectures on Special Values of L-Functions</a></em>, Perspectives in Mathematics <strong>4</strong>, Academic Press, Inc. (1988) [ISBN:978-0-12-581120-0, <a class="existingWikiWord" href="/nlab/files/EsnaultViehweg-DeligneBeilinsonCohomology.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>On holomorphic de Rham cohomology of <a class="existingWikiWord" href="/nlab/show/Stein+manifolds">Stein manifolds</a> (where it coincides with ordinary <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a>, see <a href="Stein+manifold#HolomorphicDeRhamCoincidesWithDeRhamOnSteinMfds">there</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>: <em>Quelques problemes globaux relatifs aux varietes de Stein</em>, Colloque sur les fonctions de plusieurs variables (1953) [<a href="https://doi.org/10.1007/978-3-642-39816-2_23">doi:10.1007/978-3-642-39816-2_23</a>]</p> </li> <li id="Serre54"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>: §2.1 in: <em>Cohomologie et fonctions de variables complexes</em>, Séminaire Bourbaki 2 71 (1954) [<a href="http://www.numdam.org/item/SB_1951-1954__2__213_0">numdam:SB_1951-1954__2__213_0</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 22, 2023 at 10:09:10. 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