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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 structure </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/4622/#Item_13" 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" 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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='#Idea'>Idea</a></li> <ul> <li><a href='#pure_hodge_structure'>Pure Hodge structure</a></li> <li><a href='#mixed_hodge_structure'>Mixed Hodge structure</a></li> </ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#ForAKaehlerManifold'>On the cohomology of a Kähler manifold</a></li> <li><a href='#ForAComplexAnalyticSpace'>On the cohomology of a complex analytic space</a></li> <li><a href='#OnAnAbelianGroup'>Generally on an abelian group</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> <h3 id="pure_hodge_structure">Pure Hodge structure</h3> <p>A <em>Hodge structure</em> (or <em>pure Hodge structure</em>, for emphasis) is a (bi-)<a class="existingWikiWord" href="/nlab/show/graded+object">grading</a> structure on <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> – called a <em>Hodge decomposition</em> – of the kind that is exhibited by the <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a>/complex-<a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifolds">Kähler manifolds</a>, according to the <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a>. A Hodge structure is said to be <em>of weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></em> if it behaves like the cohomology of a Kähler manifold of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>.</p> <p>If instead of considering a single <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> one considers the cohomology groups of a parameterized collection of spaces – hence the cohomology <em><a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a>/<a class="existingWikiWord" href="/nlab/show/stacks">stacks</a></em> – then one speaks of <em>variation of Hodge structure</em> (of a given weight).</p> <p>By a central theorem of Hodge theory (recalled as theorem <a class="maruku-ref" href="#HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure"></a> below) the traditional (and original) <a class="existingWikiWord" href="/nlab/show/filtered+complex">filtration</a> on the complex cohomology of a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> induced by the <a class="existingWikiWord" href="/nlab/show/harmonic+differential+forms">harmonic differential forms</a> generalizes to a <a class="existingWikiWord" href="/nlab/show/filtered+complex">filtration</a> of the complex-valued <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of any <a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a> which is simply given by the canonical degree-filtration of the <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a>.</p> <p>This means that <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> in the guise of <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> is nothing but the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> of a stage of the Hodge filtration along the “ <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> ” map from integral to complex cohomology. (A point of view highlighted for instance in <a href="#PetersSteenbrink08">Peters-Steenbrink 08, section 7.2</a>). Viewed this way Hodge structures are filtrations of stages of differential form cycle refinements of <a class="existingWikiWord" href="/nlab/show/Chern+characters">Chern characters</a> that appear in the general definition/characterization of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>, as discussed at <em><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></em> starting around the section <em><a href="differential+cohomology+diagram#DeRhamCoefficients">de Rham coefficients</a></em></p> <p>This modern point of view is also crucial for instance in the characterization of an <a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a> (see there) as the subgroup of <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a> that is in the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of the map to Hodge-filtering stage of ordinary cohomology. See at <em><a href="intermediate+Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology">intermediate Jacobian – characterization as Hodge-trivial Deligne cohomology</a></em>.</p> <h3 id="mixed_hodge_structure">Mixed Hodge structure</h3> <p>A <em>mixed Hodge structure</em> is a <a class="existingWikiWord" href="/nlab/show/filtration">filtration</a> on <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> – called a <em><a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></em> – such that the <a class="existingWikiWord" href="/nlab/show/associated+graded+object">associated graded object</a> has pure Hodge structure of weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> in each degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>. The archetypical example exhibiting this is the cohomology of <a class="existingWikiWord" href="/nlab/show/complex+varieties">complex varieties</a> that have singularities (<a href="#Deligne71">Deligne 71</a> <a href="#Deligne74">Deligne 74</a>).</p> <h2 id="definition">Definition</h2> <p>Historically, Hodge structures originate in the special structure induced on the <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham</a> <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> of a compact <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> by the existence of <a class="existingWikiWord" href="/nlab/show/harmonic+differential+forms">harmonic differential forms</a>. Below we first discuss this canonical Hodge structure</p> <ul> <li><a href="#ForAKaehlerManifold">on the cohomology of a Kähler manifold</a></li> </ul> <p>But it turns out that this Hodge structure only depends on the natural degree-<a class="existingWikiWord" href="/nlab/show/filtration">filtration</a> on the <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a> and hence more generally there is canonical Hodge structure</p> <ul> <li><a href="#ForAComplexAnalyticSpace">on the cohomology of a complex analytic space</a>.</li> </ul> <p>Abstracting from here one defines Hodges structures</p> <ul> <li><a href="#OnAnAbelianGroup">generally on abelian groups</a>.</li> </ul> <h3 id="ForAKaehlerManifold">On the cohomology of a Kähler manifold</h3> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{p,q}(X)</annotation></semantics></math> for its space of <a class="existingWikiWord" href="/nlab/show/harmonic+differential+forms">harmonic differential forms</a>, equivalently, via the <a class="existingWikiWord" href="/nlab/show/Hodge+isomorphism">Hodge isomorphism</a>, its <a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a> in bidegree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math>.</p> <p>Notice that by the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> there are canonical maps</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{p,q}(X)\to H^{p+q}(X,\mathbb{C}) </annotation></semantics></math></div> <p>to <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>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with complex <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>.</p> <div class="num_defn" id="TraditionalHodgeFiltration"> <h6 id="definition_2">Definition</h6> <p>The <em>Hodge filtration</em> on the cohomology 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> is the <a class="existingWikiWord" href="/nlab/show/filtered+complex">filtered complex</a> structure given by the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></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><munder><mo>⊕</mo><mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>≥</mo><mi>p</mi></mrow></munder><msup><mi>H</mi> <mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^p H^k(X, \mathbb{C}) \coloneqq \underset{k-q \geq p}{\oplus} H^{k-q,q}(X) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> <p>The full Hodge filtration of degree-2 cohomology is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mi>F</mi> <mn>0</mn></msup><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><msup><mi>H</mi> <mrow><mn>0</mn><mo>,</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>H</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>H</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mi>F</mi> <mn>1</mn></msup><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>H</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mi>F</mi> <mn>2</mn></msup><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} F^0 H^2(X,\mathbb{C}) & = H^{0,2}(X) \oplus H^{1,1}(X) \oplus H^{2,0}(X) \\ F^1 H^2(X,\mathbb{C}) & = \;\;\;\;\;\;\; H^{1,1}(X) \oplus H^{2,0}(X) \\ F^2 H^2(X,\mathbb{C}) & = \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; H^{2,0}(X) \end{aligned} </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>For all <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> the mid-dimensional Hodge filtration stage in even total degree is</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> <mrow><mn>2</mn><mi>p</mi></mrow></msup><mo>=</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>⋯</mi><mo>⊕</mo><msup><mi>H</mi> <mrow><mn>2</mn><mi>p</mi><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^p H^{2p} = H^{p,p}(X) \oplus H^{p+1,p-1}(X) \oplus \cdots \oplus H^{2p,0}(X) \,. </annotation></semantics></math></div></div> <h3 id="ForAComplexAnalyticSpace">On the cohomology of a complex analytic space</h3> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a>, write</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>X</mi> <mo>•</mo></msubsup><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mover><mo>⟶</mo><mo>∂</mo></mover><msubsup><mi>Ω</mi> <mi>X</mi> <mn>1</mn></msubsup><mover><mo>⟶</mo><mo>∂</mo></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>∂</mo></mover><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet_X \coloneqq (\mathcal{O}_X \stackrel{\partial}{\longrightarrow} \Omega^1_X \stackrel{\partial}{\longrightarrow} \Omega^2(X) \stackrel{\partial}{\longrightarrow} \cdots) </annotation></semantics></math></div> <p>for its <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Notice the <a href="holomorphic+de+Rham+complex#RelationToComplexCohomology">relation to complex cohomology</a> given by</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>X</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>X</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^k(X,\mathbb{C}) \simeq H^k(X,\Omega^\bullet_X) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a> 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 maxsize="1.2em" minsize="1.2em">(</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><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 maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^p \Omega^\bullet_X \coloneqq \big( 0 \to \cdots \to 0 \to \Omega^p_X \stackrel{\partial}{\longrightarrow} \Omega^{p+1} \stackrel{\partial}{\longrightarrow} \cdots \big) \,. </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 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> </div> <div class="num_defn" id="HodgeFiltrationForComplexSpace"> <h6 id="definition_4">Definition</h6> <p>The <em>Hodge filtration</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></div> <p>(e.g. <a href="#Voisin02">Voisin 02, def. 8.2</a>, <a href="#PetersSteenbrink08">Peters-Steenbrink 08, def. 2.21</a>).</p> <div class="num_theorem" id="HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure"> <h6 id="theorem">Theorem</h6> <p>When the compact <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> <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 have the structure of 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> degenerates at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math> page which implies that def. <a class="maruku-ref" href="#HodgeFiltrationForComplexSpace"></a> coincides with the traditional definition via <a class="existingWikiWord" href="/nlab/show/harmonic+differential+forms">harmonic differential forms</a>, def. <a class="maruku-ref" href="#TraditionalHodgeFiltration"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>⊕</mo><mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>≥</mo><mi>p</mi></mrow></munder><msup><mi>H</mi> <mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</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><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{k-q \geq p}{\oplus} H^{k-q,q}(X) \simeq im \left( H^k(X, F^p \Omega^\bullet_X) \to H^k(X, \Omega^\bullet_X) \right) \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#Voisin02">Voisin 02, remark 8.29</a>; <a href="#Voisin08">Voisin 08, 1.1.2</a>; <a href="#PetersSteenbrink08">Peters-Steenbrink 08, prop 2.22</a>).</p> <div class="num_remark" id="FrolicherEquivalenceInComponents"> <h6 id="remark_3">Remark</h6> <p>The equivalence in theorem <a class="maruku-ref" href="#HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure"></a> is exhibited by the following morphism.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>tot</mi><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msup><mo>,</mo><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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> tot(\Omega^{\bullet \geq p, \bullet}, \mathbf{d}= \partial + \bar \partial) </annotation></semantics></math></div> <p>for the holomorphically truncated <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, as indicated, thought of as the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of the <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault</a> <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>,</mo><mn>0</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>∂</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>∂</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>→</mo><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>∂</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>∂</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Omega^{p,0} &\stackrel{\bar \partial}{\to}& \Omega^{p-1,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \Omega^{p+1,0} &\stackrel{\bar \partial}{\to}& \Omega^{p,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \vdots && \vdots } \,. </annotation></semantics></math></div> <p>Since this is in each row the <a class="existingWikiWord" href="/nlab/show/Dolbeault+resolution">Dolbeault resolution</a> of the given sheaf of <a class="existingWikiWord" href="/nlab/show/holomorphic+differential+forms">holomorphic differential forms</a>, this total complex is indeed <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphic</a> to the (truncated) <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a>.</p> <p>The total complex is 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>k</mi></mrow><annotation encoding="application/x-tex">-k</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>⊕</mo><mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>≥</mo><mi>p</mi></mrow></munder><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>,</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\underset{k-q \geq p}{\oplus} \Omega^{k-q, q}</annotation></semantics></math> and hence globally defined closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mi>q</mi><mo>≥</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k-q \geq p,q)</annotation></semantics></math>-forms naturally inject into</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>X</mi><mo>,</mo><mi>tot</mi><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msup><mo>,</mo><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><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>tot</mi><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msup><mo>,</mo><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><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^k(X, tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial) ) \simeq H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k]) \,. </annotation></semantics></math></div> <p>Therefore given a representative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in \Omega^{p,q}_{cl}(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>α</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\alpha] \in H^{p,q}(X)</annotation></semantics></math> it is canonically sent along</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>⊕</mo><mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>≥</mo><mi>p</mi></mrow></munder><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><munder><mo>⊕</mo><mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>≥</mo><mi>p</mi></mrow></munder><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>cl</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>tot</mi><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≥</mo><mi>p</mi><mo>,</mo><mo>•</mo></mrow></msup><mo>,</mo><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><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{k-q\geq p}{\oplus} \Omega^{p,q}_{cl}(X) \simeq \underset{k-q\geq p}{\oplus} H^0(X, \Omega^{p,q}_{cl}) \to H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k]) \,. </annotation></semantics></math></div> <p>This map exhibits the equivalence in theorem <a class="maruku-ref" href="#HodgeFiltrationForComplexSpaceReproducesKaehlerHodgeStructure"></a> (e.g. <a href="#Voisin08">Voisin 08, section 1.1.2</a>).</p> <p>Dually,</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>hol</mi> <mrow><mo>≤</mo><mi>k</mi></mrow></msubsup><mo>≃</mo><mi>tot</mi><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≤</mo><mi>k</mi><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega_{hol}^{\leq k} \simeq tot( \Omega^{\bullet \leq k, \bullet}) \,. </annotation></semantics></math></div> <p>This plays a role in the discussion of <a class="existingWikiWord" href="/nlab/show/intermediate+Jacobians">intermediate Jacobians</a>, where for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dim</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim_{\mathbb{C}}(X)= k+1</annotation></semantics></math> we have</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> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msup><mi>F</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>hol</mi> <mrow><mo>•</mo><mo>≤</mo><mi>k</mi></mrow></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^{2k+1}(X,\mathbb{R}) \simeq H^{2k+1}(X,\mathbb{C}) / F^{k+1} H^{2k+1}(X,\mathbb{C}) \simeq H^{2k+1}(X, \Omega_{hol}^{\bullet \leq k}) \,. </annotation></semantics></math></div> <p>Here a real differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2k+1)</annotation></semantics></math>-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mover><mrow><msup><mi>α</mi> <mrow><mn>0</mn><mo>,</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mo>¯</mo></mover><mo>+</mo><mover><mrow><msup><mi>α</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn><mi>k</mi></mrow></msup></mrow><mo>¯</mo></mover><mo>+</mo><mi>⋯</mi><mo>+</mo><msup><mi>α</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn><mi>k</mi></mrow></msup><mo>+</mo><msup><mi>α</mi> <mrow><mn>0</mn><mo>,</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \alpha = \overline{\alpha^{0,2k+1}} + \overline{\alpha^{1, 2k}} + \cdots + \alpha^{1, 2k} + \alpha^{0,2k+1} </annotation></semantics></math></div> <p>injects via its pieces in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>≤</mo><mi>k</mi><mo>,</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>p</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>≤</mo><mi>k</mi><mo>,</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>p</mi></mrow></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>tot</mi><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo>≤</mo><mi>k</mi><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>hol</mi> <mrow><mo>•</mo><mo>≤</mo><mi>k</mi></mrow></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{p \leq k, 2k+1-p}(X) \simeq H^0(X, \Omega^{p \leq k, 2k+1-p}) \to H^0(X, tot(\Omega^{\bullet \leq k, \bullet})[-k]) \simeq H^k(\Omega_{hol}^{\bullet\leq k}) \,. </annotation></semantics></math></div></div> <h3 id="OnAnAbelianGroup">Generally on an abelian group</h3> <div class="num_defn" id="HodgeStructureOfWeightk"> <h6 id="definition_5">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℤ</mi></msub></mrow><annotation encoding="application/x-tex">H_{\mathbb{Z}}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, a <em>Hodge structure</em> of <em>weight</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{Z}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℤ</mi></msub></mrow><annotation encoding="application/x-tex">H_{\mathbb{Z}}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> decomposition of its <a class="existingWikiWord" href="/nlab/show/complexification">complexification</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℂ</mi></msub><mo>≔</mo><msub><mi>H</mi> <mi>ℤ</mi></msub><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> H_{\mathbb{C}}\coloneqq H_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{C} </annotation></semantics></math></div> <p>into <a class="existingWikiWord" href="/nlab/show/complex+vector+spaces">complex vector spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">H^{p,q}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">p +q = k</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℂ</mi></msub><mo>≃</mo><munder><mo>⊕</mo><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mi>k</mi></mrow></munder><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> H_{\mathbb{C}} \simeq \underset{p+q = k}{\oplus} H^{p,q} </annotation></semantics></math></div> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup></mrow><annotation encoding="application/x-tex">H^{q,p}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugate</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">H^{p,q}</annotation></semantics></math>:</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> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>=</mo><mover><mrow><msup><mi>H</mi> <mrow><mi>q</mi><mo>,</mo><mi>p</mi></mrow></msup></mrow><mo>¯</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^{p,q} = \overline{H^{q,p}} \,. </annotation></semantics></math></div> <p>This is an equality of the underlying sets of the complex vector spaces.</p> </div> <p>With this the above def. <a class="maruku-ref" href="#TraditionalHodgeFiltration"></a> has the following verbatim generalization</p> <div class="num_defn" id="GeneralHodgeFiltration"> <h6 id="definition_6">Definition</h6> <p>Given a Hodge structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℤ</mi></msub><mo>,</mo><mo stretchy="false">{</mo><msup><mi>H</mi> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H_{\mathbb{Z}}, \{H^{p,q}\}</annotation></semantics></math> of weight <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, def. <a class="maruku-ref" href="#HodgeStructureOfWeightk"></a>, then the associated <em>Hodge filtration</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">H_{\mathbb{C}}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/filtered+complex">filtered complex</a> structure given by the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></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><msub><mi>H</mi> <mi>ℂ</mi></msub><mo>≔</mo><munder><mo>⊕</mo><mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>≥</mo><mi>p</mi></mrow></munder><msup><mi>H</mi> <mrow><mi>k</mi><mo>−</mo><mi>q</mi><mo>,</mo><mi>q</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^p H_{\mathbb{C}} \coloneqq \underset{k-q \geq p}{\oplus} H^{k-q,q} \,. </annotation></semantics></math></div></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</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/Hodge+symmetry">Hodge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+cycle">Hodge cycle</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> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+Jacobian">intermediate Jacobian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lefschetz+decomposition">Lefschetz decomposition</a></p> </li> </ul> <h2 id="references">References</h2> <p>The basic Hodge filtration:</p> <ul> <li id="Voisin08"> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, <em>Hodge theory and the topology of compact Kähler and complex projective manifolds</em>, Lecture notes (2008) [<a href="http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Voisin-HodgeTheory.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stacks+Project">Stacks Project</a>, <em>The Hodge Filtration</em> [<a href="https://stacks.math.columbia.edu/tag/0FM7">tag:0FM7</a>]</p> </li> </ul> <p>Hodge filtration induced by the Hodge-to-de Rham spectral sequence on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>dR</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{dR}(X/S)</annotation></semantics></math> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-scheme <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> <ul> <li>The Stacks Project 50.7, tag/<a href="https://stacks.math.columbia.edu/tag/0FM7">0FM7</a></li> </ul> <p>Textbook accounts on Hodge structures:</p> <ul> <li id="Kulikov98"> <p><a class="existingWikiWord" href="/nlab/show/Valentine+S.+Kulikov">Valentine S. Kulikov</a>, <em>Mixed Hodge Structures and Singularities</em>, Cambridge University Press (1998) [<a href="https://doi.org/10.1017/CBO9780511758928">doi:10.1017/CBO9780511758928</a>]</p> </li> <li id="Voisin02"> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, section 7 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="PetersSteenbrink08"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Peters">Chris Peters</a>, <a class="existingWikiWord" href="/nlab/show/Jozef+Steenbrink">Jozef Steenbrink</a>, <em><a class="existingWikiWord" href="/nlab/show/Mixed+Hodge+Structures">Mixed Hodge Structures</a></em>, Ergebisse der Mathematik (2008) [<a href="http://www.arithgeo.ethz.ch/alpbach2012/Peters_Steenbrinck">pdf</a>, <a href="https://doi.org/10.1007/978-3-540-77017-6">doi:10.1007/978-3-540-77017-6</a>]</p> </li> </ul> <p>The notion of mixed Hodge structures was introduced in</p> <ul> <li id="Deligne71"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em>Théorie de Hodge II</em>, Publ. Math. I.H.E.S <strong>40</strong> (1971) 5-58 [<a href="http://www.numdam.org/item/PMIHES_1971__40__5_0">numdam:PMIHES_1971__40__5_0</a>, <a href="https://doi.org/10.1007/BF02684692">doi:10.1007/BF02684692</a>]</p> </li> <li id="Deligne74"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em>Théorie de Hodge III</em>, Publ. Math., I. H. E. S <strong>44</strong> (1974) 5-77 [<a href="http://www.numdam.org/item/PMIHES_1974__44__5_0">numdam:PMIHES_1974__44__5_0</a>, <a href="https://doi.org/10.1007/BF02685881">doi:10.1007/BF02685881</a>]</p> </li> </ul> <p>Review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, Section 8.4 of: <em><a class="existingWikiWord" href="/nlab/show/Noncommutative+Geometry%2C+Quantum+Fields+and+Motives">Noncommutative Geometry, Quantum Fields and Motives</a></em></li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jozef+Steenbrink">Jozef Steenbrink</a>, S. Zucker, <em>Variation of mixed Hodge structure I</em>, Invent. Math. <strong>80</strong> (1985), 489-542.</p> </li> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Hodge_structure">Hodge structure</a>,</em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Donu+Arapura">Donu Arapura</a>, <em>Mixed Hodge Structures Associated to Geometric Variations</em> [<a href="http://www.math.purdue.edu/~dvb/preprints/tifr.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eom">eom</a>,</p> <p>A.I. Ovseevich <em><a href="http://eom.springer.de/H/h047470.htm">Hodge structure</a></em>, <em><a href="http://eom.springer.de/p/p072140.htm">Period mapping</a></em>,</p> <p><a class="existingWikiWord" href="/nlab/show/Jozef+Steenbrink">Jozef Steenbrink</a>, <em><a href="http://eom.springer.de/v/v096170.htm">Variation of Hodge structure</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, <em>Hodge filtered complex bordism</em> (<a href="http://arxiv.org/abs/1212.2173">arXiv:1212.2173</a>)</p> </li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">orbifold cohomology</a>:</p> <ul> <li>Javier Fernandez, <em>Hodge structures for orbifold cohomology</em>, Proc. Amer. Math. Soc. 134 (2006), no. 9, 2511-2520 (<a href="https://arxiv.org/abs/math/0311026">arXiv:math/0311026</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 1, 2023 at 12:35:42. 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