CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> Hodge theory in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Hodge theory </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/973/#Item_4" title="Discuss this page in its dedicated thread on the nForum" 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="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</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> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</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/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> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#overview'>Overview</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#contributors'>Contributors</a></li> <li><a href='#general'>General</a></li> <li><a href='#relation_to_motives'>Relation to motives</a></li> <li><a href='#nonabelian_hodge_theory'>Nonabelian Hodge theory</a></li> <li><a href='#in_'>In <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a></a></li> </ul> </ul> </div> <h2 id="overview">Overview</h2> <p><em>Hodge theory</em> is the study of properties of (complex, rational and integer) <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> (and <a class="existingWikiWord" href="/nlab/show/motives">motives</a>) of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifolds">Kähler manifolds</a>, induced by a <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a> – given in the classical situation by <a class="existingWikiWord" href="/nlab/show/harmonic+differential+forms">harmonic differential forms</a> – and the corresponding <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a>. Hodge theory also applies in <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a>, for instance to <a class="existingWikiWord" href="/nlab/show/matroids">matroids</a> (<a href="#Huh22">Huh 22</a>).</p> <p>Central aspects of the theory include Hodge decomposition, <a class="existingWikiWord" href="/nlab/show/hard+Lefschetz+theorem">hard Lefschetz theorem</a>, <a class="existingWikiWord" href="/nlab/show/pure+Hodge+structures">pure Hodge structures</a>, <a class="existingWikiWord" href="/nlab/show/mixed+Hodge+structures">mixed Hodge structures</a>, <a class="existingWikiWord" href="/nlab/show/variations+of+Hodge+structure">variations of Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+to+de+Rham+spectral+sequence">Hodge to de Rham spectral sequence</a>, <a class="existingWikiWord" href="/nlab/show/periods">periods</a>… This basic setup is however by now vastly generalized.</p> <p>The nomenclature “nonabelian”/“noncommutative” Hodge theory may be confusing - to clarify:</p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">Nonabelian Hodge theory</a> refers to the generalization of classical Hodge theory for commutative spaces from abelian coefficients to nonabelian coefficients.</p> <p><a class="existingWikiWord" href="/nlab/show/Noncommutative+Hodge+theory">Noncommutative Hodge theory</a> refers to the generalization of classical Hodge theory from commutative geometry (e.g. algebraic varieties) to <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a> (e.g. <a class="existingWikiWord" href="/nlab/show/dg-categories">dg-categories</a> – see also the page on <a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a>). The coefficients in noncommutative Hodge theory are abelian.</p> <p>Ultimately, the goal is to develop a Hodge theory for noncommutative spaces with nonabelian coefficients.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/p-adic+Hodge+theory">p-adic Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/harmonic+differential+form">harmonic differential form</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+conjecture">Hodge conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Lefschetz+hyperplane+theorem">Lefschetz hyperplane theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motive">motive</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="contributors">Contributors</h3> <p>The main historical contributors to Hodge theory include <a class="existingWikiWord" href="/nlab/show/William+Hodge">William Hodge</a>, <a class="existingWikiWord" href="/nlab/show/Phillip+Griffiths">Phillip Griffiths</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>. More recently a noncommutative generalization and refinement of Hodge theory is emerging in the work of <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">M. Kontsevich</a> and collaborators (D. Kaledin, L. Katzarkov, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>…) with precursors in the works of <a class="existingWikiWord" href="/nlab/show/Carlos+Simpson">Carlos Simpson</a>, <a class="existingWikiWord" href="/nlab/show/Mikio+Sato">Mikio Sato</a>, <a class="existingWikiWord" href="/nlab/show/Claus+Hertling">Claus Hertling</a> and others. Another very interesting and complex picture is developing for a number of years in works of Goncharov.</p> <h3 id="general">General</h3> <ul id="GreenVoisinMurre"> <li> <p>Bertin, Demailly, <a class="existingWikiWord" href="/nlab/show/Luc+Illusie">Luc Illusie</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Peters">Chris Peters</a>, <em>Introduction to Hodge theory</em>, AMS (1996)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, <em>Hodge theory and the topology of complex Kähler and complex projective manifolds</em> (survey, <a href="http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf">pdf</a>)</p> </li> <li id="Voisin02"> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, <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) [<a href="https://doi.org/10.1017/CBO9780511615344">doi:10.1017/CBO9780511615344</a>]</p> </li> <li> <p>Mark Green, <a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, Jacob Murre, <em>Algebraic cycles and Hodge theory</em>, Lecture Notes in Mathematics, 1594 (1993)</p> </li> </ul> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Phillip+Griffiths">Phillip Griffiths</a>, <em>Periods of integrals on algebraic manifolds I,II,III</em>, Amer. J. Math. <strong>90</strong>, 568–626, 808–865 (1968) ; Publ. Math. de l’IHÉS <strong>38</strong>, 228–296 (1970) <a href="http://www.numdam.org/item?id=PMIHES_1970__38__125_0">numdam</a></p> </li> <li> <p>P. Griffiths, <em>Hodge theory - from Abel to Deligne</em>, talk, Inst. Adv. Studies 2016 <a href="https://www.youtube.com/watch?v=hPyDz5R5YaY">yt</a>; <em>Some geometric applications of Hodge theory</em>, Oct 2020, <a href="https://www.youtube.com/watch?v=MA8FhSY3FlU">yt</a></p> </li> <li> <p>M. Saito, <em>Mixed Hodge modules</em>, Publ. R.I.M.S. Kyoto Univ. <strong>26</strong> (1990) pp. 221–333.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L.+Katzarkov">L. Katzarkov</a>, <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">M. Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, <em>Hodge theoretic aspects of mirror symmetry</em>, <a href="http://arxiv.org/abs/0806.0107">arxiv/0806.0107</a> Proc. of Symposia in Pure Math. <strong>78</strong> (2008), “From Hodge theory to integrability and TQFT: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>tt</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">tt^*</annotation></semantics></math>-geometry”, eds. Ron Y. Donagi and Katrin Wendland, 87-174</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D.+Kaledin">D. Kaledin</a>, <em>Cartier isomorphism and Hodge theory in the non-commutative case</em>, Arithmetic geometry, 537–562, Clay Math. Proc. <strong>8</strong>, Amer. Math. Soc. 2009, <a href="https://arxiv.org/abs/0708.1574">arxiv/0708.1574</a></p> </li> <li> <p>D. Kaledin, <em>Tokyo lectures “Homological methods in non-commutative geometry”</em>, <a href="http://imperium.lenin.ru/~kaledin/tokyo/final.pdf">pdf</a>, <a href="http://imperium.lenin.ru/~kaledin/tokyo/final.tex">TeX</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Claus+Hertling">Claus Hertling</a>, Christian Sevenheck, <em>Twistor structures, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>tt</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">tt^*</annotation></semantics></math>-geometry and singularity theory</em>, pp. 49–74 in: “From tQFT to tt* and integrability”, Proceedings of Symposia in Pure Mathematics <strong>78</strong> (2007, Augsburg, Germany), AMS 2008; <a href="https://arxiv.org/abs/0807.2199">arxiv:0807.2199</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Claus+Hertling">Claus Hertling</a>, <a class="existingWikiWord" href="/nlab/show/Claude+Sabbah">Claude Sabbah</a>, <em>Examples of non-commutative Hodge structure</em> (v1 title: Fourier-Laplace transform of flat unitary connections and TERP structures), <a href="https://arxiv.org/abs/0912.2754">arxiv/0912.2754</a></p> </li> <li> <p>A. B. Goncharov, <em>Hodge correlators</em>, <a href="http://arxiv.org/abs/0803.0297">arxiv/0803.0297</a>, <em>Hodge correlators II</em>, <a href="http://arxiv.org/abs/0807.4855">arxiv/0807.4855</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C.+T.+C.+Wall">C. T. C. Wall</a>, <em>Periods of integrals and topology of algebraic varieties</em>, <a href="http://www.jstor.org/stable/2397500">jstor</a> A course in Hodge theory : with emphasis on multiple integrals</p> </li> <li> <p>Hossein Movasati, <em>A course in Hodge theory: with emphasis on multiple integrals</em>, Somerville: International Press of Boston 2021.</p> </li> </ul> <h3 id="relation_to_motives">Relation to motives</h3> <p>Discussions of Hodge theory in relation to <a class="existingWikiWord" href="/nlab/show/motives">motives</a> can be found in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Peters">Chris Peters</a>, <em>Tata Lectures on Motivic Aspects of Hodge Theory</em> (<a href="http://www-fourier.ujf-grenoble.fr/~peters/Books/motivic.f/Tatalects.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chris+Peters">Chris Peters</a>, <em>Lectures on Motivic Aspects of Hodge Theory: The Hodge Characteristic.</em>, Summer School Istanbul 2014, <a href="http://www-fourier.ujf-grenoble.fr/~peters/Books/motivic.f/MotAspects.pdf">pdf</a></p> </li> </ul> <h3 id="nonabelian_hodge_theory">Nonabelian Hodge theory</h3> <p>General references on nonabelian Hodge theory include:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ron+Donagi">Ron Donagi</a>, <a class="existingWikiWord" href="/nlab/show/Tony+Pantev">Tony Pantev</a>, <em>Lectures on the geometric Langlands conjecture and non-abelian Hodge theory</em>, 2009 (<a href="http://www.icmat.es/seminarios/langlands/school/handouts/pantev.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alberto+Garc%C3%ADa+Raboso">Alberto García Raboso</a>, <a class="existingWikiWord" href="/nlab/show/Steven+Rayan">Steven Rayan</a>, <em>Introduction to Nonabelian Hodge Theory: flat connections, Higgs bundles, and complex variations of Hodge structure</em>, Fields Inst. Monogr. 34 (2015), 131–171 (<a href="https://arxiv.org/abs/1406.1693">arXiv</a>) (<a href="http://link.springer.com/chapter/10.1007/978-1-4939-2830-9_5">Springer</a>)</p> </li> </ul> <h3 id="in_">In <a class="existingWikiWord" href="/nlab/show/combinatorics">combinatorics</a></h3> <ul> <li id="Huh22"><a class="existingWikiWord" href="/nlab/show/June+Huh">June Huh</a>, <em>Combinatorics and Hodge theory</em>, Proc. Int. Cong. Math. <strong>1</strong> (2022) <p>[<a href="">doi:10.4171/ICM2022/205</a>, <a href="https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/jh.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Huh-CombinatoricsAndHodgeTheory.pdf" title="pdf">pdf</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 18, 2023 at 05:47:00. See the <a href="/nlab/history/Hodge+theory" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/Hodge+theory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/973/#Item_4">Discuss</a><span class="backintime"><a href="/nlab/revision/Hodge+theory/28" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Hodge+theory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Hodge+theory" accesskey="S" class="navlink" id="history" rel="nofollow">History (28 revisions)</a> <a href="/nlab/show/Hodge+theory/cite" style="color: black">Cite</a> <a href="/nlab/print/Hodge+theory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Hodge+theory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>