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Kähler manifold in nLab
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<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 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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="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</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+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </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> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#LinearKählerStructure'>Linear Kähler structure</a></li> <li><a href='#khler_manifolds'>Kähler manifolds</a></li> <li><a href='#InTermsOfGStructure'>In terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_almost_complex_manifold'>Relation to (almost) complex manifold</a></li> <li><a href='#relation_to_symplectic_manifolds'>Relation to symplectic manifolds</a></li> <li><a href='#relation_to_spinstructures'>Relation to Spin-structures</a></li> <li><a href='#HodgeStarOperator'>Hodge star operator</a></li> <li><a href='#HodgeStructure'>Hodge structure</a></li> <li><a href='#as_riemannian_manifolds'>As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>-Riemannian manifolds</a></li> </ul> <li><a href='#kapranov_algebras'>Kapranov <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_{\infty}</annotation></semantics></math>-algebras</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Kähler manifold</em> is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> compatibly equipped with</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+structure">Riemannian structure</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+structure">symplectic structure</a>.</p> </li> </ol> <p>If the symplectic structure is not compatibly present, it is just a <a class="existingWikiWord" href="/nlab/show/Hermitian+manifold">Hermitian manifold</a>.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></th><th>+ <a class="existingWikiWord" href="/nlab/show/Riemannian+structure">Riemannian structure</a></th><th>+ <a class="existingWikiWord" href="/nlab/show/symplectic+structure">symplectic structure</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hermitian+structure">Hermitian structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+structure">Kähler structure</a></td></tr> </tbody></table> <p>Where a Riemannian manifold is a real <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> equipped with a nondegenerate smooth symmetric 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>), an <strong>almost Kähler manifold</strong> is a <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex holomorphic manifold</a> equipped with a nondegenerate hermitian 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> (the <strong>Kähler <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form</strong>). The real <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> is replaced with the complex cotangent bundle, and symmetry is replaced with hermitian symmetry. An almost Kähler manifold is a <strong>Kähler manifold</strong> if it satisfies an additional integrability condition.</p> <p>The Kähler 2-form can be decomposed as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>g</mi><mo>+</mo><mi>i</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">h = g+i\omega</annotation></semantics></math>; here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>.</p> <h2 id="definition">Definition</h2> <h3 id="LinearKählerStructure">Linear Kähler structure</h3> <div class="num_defn" id="KaehlerVectorSpace"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a>. Then a <em>linear Kähler structure</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is</p> <ol> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/linear+complex+structure">linear complex structure</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, namely a <a class="existingWikiWord" href="/nlab/show/linear+map">linear</a> <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>J</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> J \;\colon\; V \to V </annotation></semantics></math></div> <p>whose <a class="existingWikiWord" href="/nlab/show/composition">composition</a> with itself is minus the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>∘</mo><mi>J</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>id</mi> <mi>V</mi></msub></mrow><annotation encoding="application/x-tex"> J \circ J = - id_V </annotation></semantics></math></div></li> <li> <p>a skew-symmetric <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mo>∧</mo> <mn>2</mn></msup><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \omega \in \wedge^2 V^\ast </annotation></semantics></math></div></li> </ol> <p>such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega(J(-),J(-)) = \omega(-,-)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≔</mo><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(-,-) \coloneqq \omega(-,J(-))</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, namely</p> <p>a non-degenerate positive-definite <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></p> <p>(necessarily symmetric, due to the other properties: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(w,v) = \omega(w,J(v)) = -\omega(J(v),w) = - \omega(J(J(v)), J(w)) = +\omega(v,J(w)) = g(v,w)</annotation></semantics></math>).</p> </li> </ol> </div> <p>(e.g. <a href="#Boalch09">Boalch 09, p. 26-27</a>)</p> <p>Linear Kähler space structure may conveniently be encoded in terms of <a class="existingWikiWord" href="/nlab/show/Hermitian+space">Hermitian space</a> structure:</p> <div class="num_defn" id="HermitianForm"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a> and <a class="existingWikiWord" href="/nlab/show/Hermitian+space">Hermitian space</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> equipped with a <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">J\colon V \to V</annotation></semantics></math>. Then a <em><a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is</p> <ul> <li> <p>a complex-valued real-<a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>⟶</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> h \;\colon\; V \otimes V \longrightarrow \mathbb{C} </annotation></semantics></math></div></li> </ul> <p>such that this is <em>symmetric sesquilinear</em>, in that:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is complex-linear in the first argument;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo>(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">h(w,v) = \left(h(v,w) \right)^\ast</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v,w \in V</annotation></semantics></math></p> </li> </ol> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(-)^\ast</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/complex+conjugation">complex conjugation</a>.</p> <p>A Hermitian form is <em>positive definite</em> (often assumed by default) if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v \in V</annotation></semantics></math></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h(v,v) \geq 0</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mphantom><mi>AA</mi></mphantom><mo>⇔</mo><mphantom><mi>AA</mi></mphantom><mi>v</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0</annotation></semantics></math>.</p> </li> </ol> <p>A <a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,J)</annotation></semantics></math> equipped with a (positive definite) Hermitian form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is called a (positive definite) <em><a class="existingWikiWord" href="/nlab/show/Hermitian+space">Hermitian space</a></em>.</p> </div> <div class="num_prop" id="BasicPropertiesOfHermitianForms"> <h6 id="proposition">Proposition</h6> <p><strong>(basic properties of <a class="existingWikiWord" href="/nlab/show/Hermitian+forms">Hermitian forms</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">((V,J),h)</annotation></semantics></math> be a positive definite <a class="existingWikiWord" href="/nlab/show/Hermitian+space">Hermitian space</a> (def. <a class="maruku-ref" href="#HermitianForm"></a>). Then</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/real+part">real part</a> of the <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Re</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> g(-,-) \;\coloneqq\; Re(h(-,-)) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, hence a symmetric positive-definite real-<a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> g \;\colon\; V \otimes V \to \mathbb{R} </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/imaginary+part">imaginary part</a> of the <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega(-,-) \;\coloneqq\; -Im(h(-,-)) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, hence a non-degenerate skew-symmetric real-<a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>∧</mo><mi>V</mi><mo>→</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega \;\colon\; V \wedge V \to \mathbb{R} \,. </annotation></semantics></math></div></li> </ol> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>g</mi><mo>−</mo><mi>i</mi><mi>ω</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> h = g - i \omega \,. </annotation></semantics></math></div> <p>The two components are related by</p> <div class="maruku-equation" id="eq:RelationBetweennOmegaAndgOnHermitianSpace"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mphantom><mi>AAAAA</mi></mphantom><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega(v,w) \;=\; g(J(v),w) \phantom{AAAAA} g(v,w) \;=\; \omega(v,J(v)) \,. </annotation></semantics></math></div> <p>Finally</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> h(J(-),J(-)) = h(-,-) </annotation></semantics></math></div> <p>and so the Riemannian metrics <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> appearing from (and fully determining) Hermitian forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>g</mi><mo>−</mo><mi>i</mi><mi>ω</mi></mrow><annotation encoding="application/x-tex">h = g - i \omega</annotation></semantics></math> are precisely those for which</p> <div class="maruku-equation" id="eq:HermitianMetric"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g(J(-),J(-)) = g(-,-) \,. </annotation></semantics></math></div> <p>These are called the <em><a class="existingWikiWord" href="/nlab/show/Hermitian+metrics">Hermitian metrics</a></em>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The positive-definiteness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is immediate from that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>. The symmetry of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> follows from the symmetric sesquilinearity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math>:</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><mi>g</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mi>Re</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Re</mi><mrow><mo>(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Re</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} g(w,v) & \coloneqq Re(h(w,v)) \\ & = Re\left( h(v,w)^\ast\right) \\ & = Re(h(v,w)) \\ & = g(v,w) \,. \end{aligned} </annotation></semantics></math></div> <p>That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is invariant under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> follows from its sesquilinarity</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><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>i</mi><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} h(J(v),J(w)) &= i h(v,J(w)) \\ & = i (h(J(w),v))^\ast \\ & = i (-i) (h(w,v))^\ast \\ & = h(v,w) \end{aligned} </annotation></semantics></math></div> <p>and this immediately implies the corresponding invariance of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>.</p> <p>Analogously it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is skew symmetric:</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><mi>ω</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Im</mi><mrow><mo>(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \omega(w,v) & \coloneqq -Im(h(w,v)) \\ & = -Im\left( h(v,w)^\ast\right) \\ & = Im(h(v,w)) \\ & = - \omega(v,w) \,, \end{aligned} </annotation></semantics></math></div> <p>and the relation between the two components:</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><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Re</mi><mo stretchy="false">(</mo><mi>i</mi><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Re</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \omega(v,w) & = - Im(h(v,w)) \\ & = Re(i h(v,w)) \\ & = Re(h(J(v),w)) \\ & = g(J(v),w) \end{aligned} </annotation></semantics></math></div> <p>as well as</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><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>Re</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>i</mi><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><msup><mi>J</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>Im</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} g(v,w) & = Re(h(v,w) \\ & = Im(i h(v,w)) \\ & = Im(h(J(v),w)) \\ & = Im(h(J^2(v),J(w))) \\ & = - Im(h(v,J(w))) \\ & = \omega(v,J(w)) \,. \end{aligned} </annotation></semantics></math></div></div> <p>As a corollary:</p> <div class="num_prop" id="RelationBetweenKaehlerVectorSpacesAndHermitianSpaces"> <h6 id="proposition_2">Proposition</h6> <p><strong>(relation between <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+spaces">Kähler vector spaces</a> and <a class="existingWikiWord" href="/nlab/show/Hermitian+spaces">Hermitian spaces</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/linear+complex+structure">linear complex structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>, then the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mo>∧</mo> <mn>2</mn></msup><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\omega \in \wedge^2 V^\ast</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/linear+K%C3%A4hler+structure">linear Kähler structure</a> (def. <a class="maruku-ref" href="#KaehlerVectorSpace"></a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">g \in V \otimes V \to \mathbb{R}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Hermitian+metric">Hermitian metric</a> <a class="maruku-eqref" href="#eq:HermitianMetric">(2)</a></p> </li> </ol> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> are related by <a class="maruku-eqref" href="#eq:RelationBetweennOmegaAndgOnHermitianSpace">(1)</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mphantom><mi>AAAAA</mi></mphantom><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega(v,w) \;=\; g(J(v),w) \phantom{AAAAA} g(v,w) \;=\; \omega(v,J(w)) \,. </annotation></semantics></math></div></div> <h3 id="khler_manifolds">Kähler manifolds</h3> <p>(…)</p> <h3 id="InTermsOfGStructure">In terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure</h3> <p>A Kähler manifold is a <a class="existingWikiWord" href="/nlab/show/integrability+of+G-structure">first-order integrable</a> <a class="existingWikiWord" href="/nlab/show/almost+Hermitian+structure">almost Hermitian structure</a>, hence a first order integrable <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G = U(n) \hookrightarrow GL(2n,\mathbb{R})</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a>.</p> <p>By the fact (see at <em><a href="unitary+group#RelationToOrthogonalSymplecticAndGeneralLinearGroup">unitary group – relation to orthogonal, symplectic and general linear group</a></em>) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow></munder><mi>Sp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>GL</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow></munder><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})</annotation></semantics></math> this means that a Kähler manifold structure is precisely a joint <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a>/<a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> structure, <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> structure and <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> structure.</p> <p>(e.g. <a href="#Moroianu07">Moroianu 07, 11.1</a>, <a href="#Verbitsky09">Verbitsky 09</a>)</p> <h2 id="Examples">Examples</h2> <p>The archetypical elementary example is the following:</p> <div class="num_example" id="StandardAlmostKaehlerVectorSpaces"> <h6 id="example">Example</h6> <p><strong>(standard <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>≔</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">V \coloneqq \mathbb{R}^2</annotation></semantics></math> be the 2-dimensional <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> equipped with the <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> which is given by the canonical identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>≃</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^2 \simeq \mathbb{C}</annotation></semantics></math>, hence, in terms of the canonical <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e_i)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>, this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>J</mi> <mi>i</mi></msup><msub><mrow></mrow> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> J = (J^i{}_j) \coloneqq \left( \array{ 0 & -1 \\ 1 & 0 } \right) \,. </annotation></semantics></math></div> <p>Moreover let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>ω</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \omega = (\omega_{i j}) \coloneqq \left( \array{0 & 1 \\ -1 & 0} \right) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g = (g_{i j}) \coloneqq \left( \array{ 1 & 0 \\ 0 & 1} \right) \,. </annotation></semantics></math></div> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>J</mi><mo>,</mo><mi>ω</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V, J, \omega, g)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+vector+space">Kähler vector space</a> (def. <a class="maruku-ref" href="#KaehlerVectorSpace"></a>)</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> in the standard way and equipped with the <a class="existingWikiWord" href="/nlab/show/bilinear+forms">bilinear forms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>,</mo><mi>ω</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">J, \omega g</annotation></semantics></math> extended as constant rank-2 <a class="existingWikiWord" href="/nlab/show/tensors">tensors</a> over this manifold.</p> <p>If we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R} </annotation></semantics></math></div> <p>for the standard <a class="existingWikiWord" href="/nlab/show/coordinate+functions">coordinate functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>≔</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>z</mi><mo>¯</mo></mover><mo>≔</mo><mi>x</mi><mo>−</mo><mi>i</mi><mi>y</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> \overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C} </annotation></semantics></math></div> <p>for the corresponding complex coordinates, then this translates to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega \in \Omega^2(\mathbb{R}^2) </annotation></semantics></math></div> <p>being the <a class="existingWikiWord" href="/nlab/show/differential+2-form">differential 2-form</a> given by</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><mi>ω</mi></mtd> <mtd><mo>=</mo><mi>d</mi><mi>x</mi><mo>∧</mo><mi>d</mi><mi>y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mn>2</mn></mfrac></mstyle><mi>d</mi><mi>z</mi><mo>∧</mo><mi>d</mi><mover><mi>z</mi><mo>¯</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \omega & = d x \wedge d y \\ & = \tfrac{i}{2} d z \wedge d \overline{z} \end{aligned} </annotation></semantics></math></div> <p>and with <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> <a class="existingWikiWord" href="/nlab/show/tensor">tensor</a> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>=</mo><mi>d</mi><mi>x</mi><mo>⊗</mo><mi>d</mi><mi>x</mi><mo>+</mo><mi>d</mi><mi>y</mi><mo>⊗</mo><mi>d</mi><mi>y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g = d x \otimes d x + d y \otimes d y \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a> is given by</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><mi>h</mi></mtd> <mtd><mo>=</mo><mi>g</mi><mo>−</mo><mi>i</mi><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>d</mi><mi>z</mi><mo>⊗</mo><mi>d</mi><mover><mi>z</mi><mo>¯</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} h & = g - i \omega \\ & = d z \otimes d \overline{z} \,. \end{aligned} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is elementary, but, for the record, here is one way to make it fully explicit (we use <a class="existingWikiWord" href="/nlab/show/Einstein+summation+convention">Einstein summation convention</a> and “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot</annotation></semantics></math>” denotes <a class="existingWikiWord" href="/nlab/show/matrix+multiplication">matrix multiplication</a>):</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><msub><mi>ω</mi> <mrow><mi>i</mi><mi>j</mi><mo>′</mo></mrow></msub><msup><mi>J</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msup><msub><mrow></mrow> <mi>j</mi></msub></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \omega_{i j'} J^{j'}{}_j & = \left( \array{ 0 & 1 \\ -1 & 0 } \right) \cdot \left( \array{ 0 & -1 \\ 1 & 0 } \right) \\ & = \left( \array{ 1 & 0 \\ 0 & 1 } \right) \\ & = g_{i j} \end{aligned} </annotation></semantics></math></div> <p>and similarly</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><mi>ω</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>J</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd> <mtd><mo>=</mo><msub><mi>ω</mi> <mrow><mi>i</mi><mo>′</mo><mi>j</mi><mo>′</mo></mrow></msub><msup><mi>J</mi> <mrow><mi>i</mi><mo>′</mo></mrow></msup><msub><mrow></mrow> <mi>i</mi></msub><msup><mi>J</mi> <mrow><mi>j</mi><mo>′</mo></mrow></msup><msub><mrow></mrow> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msup><mi>J</mi> <mi>t</mi></msup><mo>⋅</mo><mi>ω</mi><mo>⋅</mo><mi>J</mi><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mrow><mo>(</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>)</mo></mrow> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mrow><mo>(</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>)</mo></mrow> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>ω</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \omega(J(-),J(-))_{i j} & = \omega_{i' j'} J^{i'}{}_{i} J^{j'}{}_{j} \\ & = (J^t \cdot \omega \cdot J)_{i j} \\ & = \left( \left( \array{ 0 & 1 \\ -1 & 0 } \right) \cdot \left( \array{ 0 & 1 \\ -1 & 0 } \right) \cdot \left( \array{ 0 & -1 \\ 1 & 0 } \right) \right)_{i j} \\ & = \left( \left( \array{ -1 & 0 \\ 0 & -1 } \right) \cdot \left( \array{ 0 & -1 \\ 1 & 0 } \right) \right)_{i j} \\ & = \left( \array{ 0 & 1 \\ -1 & 0 } \right)_{i j} \\ & = \omega_{i j} \end{aligned} </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Fubini-Study+metric">Fubini-Study metric</a>)</strong></p> <p>There is a unique (up to a scalar) hermitian metric on <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a> (which may be normalized), the <em><a class="existingWikiWord" href="/nlab/show/Fubini-Study+metric">Fubini-Study metric</a></em>.</p> <p>All analytic subvarieties of a complex projective space are in fact <a class="existingWikiWord" href="/nlab/show/algebraic+variety">algebraic subvarieties</a> and they inherit the Kähler structure from the projective space.</p> <p>Examples include <a class="existingWikiWord" href="/nlab/show/complex+tori">complex tori</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}^n/L</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a lattice in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^n</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/K3-surfaces">K3-surfaces</a>, compact <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a>, quadrics, products of projective spaces and so on.</p> </div> <h2 id="properties">Properties</h2> <h3 id="relation_to_almost_complex_manifold">Relation to (almost) complex manifold</h3> <blockquote> <p>The following based on <a href="http://mathoverflow.net/a/73330/381">this MO comment</a> by <a class="existingWikiWord" href="/nlab/show/Spiro+Karigiannis">Spiro Karigiannis</a></p> </blockquote> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>J</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, J)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/almost+complex+manifold">almost complex manifold</a>, then there is a notion of smooth complex-valued <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> of type <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>. A complex valued <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math> precisely if it satisfies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>J</mi><mi>v</mi><mo>,</mo><mi>J</mi><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega(J v,J w) = \omega(v,w) </annotation></semantics></math></div> <p>for all smooth <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>,</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">v,w</annotation></semantics></math> on <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>. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is a <em>real</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form of type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ω</mi><mo>¯</mo></mover><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">\overline \omega = \omega</annotation></semantics></math>. Setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>J</mi><mi>w</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> g(v,w) = \omega(v, J w), </annotation></semantics></math></div> <p>defines a smooth symmetric rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/tensor+field">tensor field</a>. This is a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> precisely if it is fiberwise a <a class="existingWikiWord" href="/nlab/show/positive+definite+bilinear+form">positive definite bilinear form</a>. If it <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>J</mi><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(-,-) = \omega(-,J -)</annotation></semantics></math> is hence a Riemannian metric, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega(-,-)</annotation></semantics></math> is called positive definite, too.</p> <p>The triple of data <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>J</mi><mo>,</mo><mi>ω</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(J, \omega, g)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/almost+complex+structure">almost complex structure</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is a real positive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is the associated <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> this way define an <em><a class="existingWikiWord" href="/nlab/show/almost+Hermitian+manifold">almost Hermitian manifold</a></em>.</p> <p>Now the condition 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> to be a Kähler is that <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/complex+manifold">complex manifold</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> is integrable) and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>ω</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d\omega = 0</annotation></semantics></math>. Equivalently that for the <a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mi>ω</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla \omega = 0</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mi>J</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla J = 0</annotation></semantics></math>.</p> <p>Hence given a <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>, together with a <em>closed</em> real <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, the only additional condition required to ensure that it defines a Kähler metric is that it be a positive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,1)</annotation></semantics></math>-form.</p> <h3 id="relation_to_symplectic_manifolds">Relation to symplectic manifolds</h3> <p>Lifting a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> structure to a Kähler manifold structure is also called choosing a <em><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a></em>.</p> <h3 id="relation_to_spinstructures">Relation to Spin-structures</h3> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>A spin structure on a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/Hermitian+manifold">Hermitian manifold</a> (<a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler 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> of complex <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>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> exists precisely if, equivalently</p> <ul> <li> <p>there is a choice of <a class="existingWikiWord" href="/nlab/show/square+root">square root</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\sqrt{\Omega^{n,0}}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Omega^{n,0}</annotation></semantics></math> (a “<a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a>”);</p> </li> <li> <p>there is a trivialization of the <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_1(T X)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>.</p> </li> </ul> </div> <p>In this case one has:</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub><mo>≃</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msubsup><mo>⊗</mo><msub><msqrt><mrow><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msup></mrow></msqrt> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex"> S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">S_X</annotation></semantics></math> on <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 the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of the <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a> with the corresponding <a class="existingWikiWord" href="/nlab/show/Theta+characteristic">Theta characteristic</a>;</p> <p>Moreover, the corresponding <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> is the <a class="existingWikiWord" href="/nlab/show/Dolbeault-Dirac+operator">Dolbeault-Dirac operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∂</mo><mo>¯</mo></mover><mo>+</mo><msup><mover><mo>∂</mo><mo>¯</mo></mover> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\overline{\partial} + \overline{\partial}^\ast</annotation></semantics></math>.</p> </div> <p>This is due to (<a href="#Hitchin74">Hitchin 74</a>). A textbook account is for instance in (<a href="#Friedrich74">Friedrich 74, around p. 79 and p. 82</a>).</p> <h3 id="HodgeStarOperator">Hodge star operator</h3> <p>On a Kähler manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> 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><msub><mi>dim</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">dim_{\mathbb{C}}(\Sigma) = n</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Hodge+star+operator">Hodge star operator</a> acts on the <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⋆</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</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>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</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"> \star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X) \,. </annotation></semantics></math></div> <p>(notice the exchange of the role of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>) See e.g. (<a href="#BiquerdHoering08">BiquerdHöring 08, p. 79</a>).</p> <h3 id="HodgeStructure">Hodge structure</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a> asserts that for a compact Kähler manifold, the canonical <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>-grading of its <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> descends to its <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a>/<a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>. The resulting structure is called a <em><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></em>, and is indeed the archetypical example of such.</p> <h3 id="as_riemannian_manifolds">As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>-Riemannian manifolds</h3> <div> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>𝔸</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{A}\;</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math>Riemannian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi></mrow><annotation encoding="application/x-tex">\mathbb{A}</annotation></semantics></math>-manifolds<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math>special Riemannian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔸</mi></mrow><annotation encoding="application/x-tex">\mathbb{A}</annotation></semantics></math>-manifolds<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{R}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>ℂ</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{C}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>ℍ</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{H}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/quaternion-K%C3%A4hler+manifold">quaternion-Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/hyperk%C3%A4hler+manifold">hyperkähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/octonions">octonions</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mi>𝕆</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\mathbb{O}\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29-manifold">Spin(7)-manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math></td></tr> </tbody></table> <p>(<a href="special+holonomy#Leung02">Leung 02</a>)</p> </div> <h2 id="kapranov_algebras">Kapranov <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_{\infty}</annotation></semantics></math>-algebras</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> further denote the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of a <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mrow><msub><mi>T</mi> <mi>X</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\alpha_{T_X}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah class</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</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>.<br />One of the main observations in <a href="#Kapranov99">Kapranov 1999</a> is the following (Theorem 2.3. <a href="#Kapranov99">Kapranov 1999</a>):</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p></p> <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 any <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a> of commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math>-algebras. Then:</p> <p>(a) The maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>H</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^i(X, T \otimes A) \otimes H^j(X, T \otimes A) \longrightarrow H^{i+j+1}(X, T \otimes A)</annotation></semantics></math> <br />given by composing the <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mrow><msub><mi>T</mi> <mi>X</mi></msub></mrow></msub><mo>∈</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi mathvariant="normal">Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mi>T</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_{T_X} \in H^1(X, \mathrm{Hom}(S^2 T, T))</annotation></semantics></math>, make the <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{\bullet - 1}(X, T \otimes A)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>.</p> <p>(b) For any <a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> on <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> the maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>H</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^i(X, T \otimes A) \otimes H^j(X, E \otimes A) \to H^{i+j+1}(X, E \otimes A)</annotation></semantics></math> given by composing the <a class="existingWikiWord" href="/nlab/show/cup-product">cup-product</a> with the <a class="existingWikiWord" href="/nlab/show/Atiyah+class">Atiyah class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>E</mi></msub><mo>∈</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi mathvariant="normal">Hom</mi><mo stretchy="false">(</mo><mi>T</mi><mo>⊗</mo><mi>E</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\alpha_E \in H^1(X, \mathrm{Hom}(T \otimes E, E)),</annotation></semantics></math> make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{\bullet - 1}(X, E \otimes A)</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/graded+module">graded</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>T</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{\bullet - 1}(X, T \otimes A)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>.</p> <p></p> </div> </p> <p>Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a Kähler manifold then this graded Lie algebra admits a lift to a <a class="existingWikiWord" href="/nlab/show/L-infinity-algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebra</a> on the shifted <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>.</p> <p>Given a <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+metric">Kähler metric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/connection">connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo>˜</mo></mover><mo>=</mo><mo>∇</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mover><mo>∂</mo><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{\nabla} = \nabla + \overline{\partial}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∂</mo><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\partial}</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>-connection defining the <a class="existingWikiWord" href="/nlab/show/complex+structure">complex manifold</a>. Then the <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{\nabla}</annotation></semantics></math> is the Dolbeault representative of the <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mrow><msub><mi>T</mi> <mi>X</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\alpha_{T_X}</annotation></semantics></math>. In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> being Kähler the connection is torsionless and the curvature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi mathvariant="normal">Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mi>T</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \in \Omega^{0,1}(\mathrm{Hom}(S^2 T, T))</annotation></semantics></math>.</p> <p>Further, the author defines <a class="existingWikiWord" href="/nlab/show/tensor+fields">tensor fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">R_n</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math>, as higher <a class="existingWikiWord" href="/nlab/show/covariant+derivatives">covariant derivatives</a> of the curvature:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mspace width="1em"></mspace><msub><mi>R</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi mathvariant="normal">Hom</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mi>T</mi><mo>⊗</mo><msup><mi>T</mi> <mrow><mo>⊗</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mspace width="1em"></mspace><msub><mi>R</mi> <mn>2</mn></msub><mo>≔</mo><mi>R</mi><mo>,</mo><mspace width="1em"></mspace><msub><mi>R</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>∇</mo><msub><mi>R</mi> <mi>i</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> \quad R_n \,\in\, \Omega^{0,1}\big( \mathrm{Hom}(S^2 T \otimes T^{\otimes (n-2)}, T) \big), \quad R_2 \coloneqq R, \quad R_{i+1} = \nabla R_i. </annotation></semantics></math></div> <p> <div class='num_theorem' id='KapranovTheorem'> <h6>Theorem</h6> <p></p> <p>The maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mi>n</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><msub><mi>j</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><msub><mi>j</mi> <mi>n</mi></msub></mrow></msup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><msub><mi>j</mi> <mn>1</mn></msub><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>j</mi> <mi>n</mi></msub><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="1em"></mspace><mi>n</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow><annotation encoding="application/x-tex"> b_n \,\colon\, \Omega^{0, j_1}(T) \otimes \cdots \otimes \Omega^{0, j_n}(T) \longrightarrow \Omega^{0, j_1 + \cdots + j_n + 1}(T) \,, \quad n \geq 2, </annotation></semantics></math></div> <p>given by composing the <a class="existingWikiWord" href="/nlab/show/wedge+product">wedge product</a> (with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo></mrow></msup><mo stretchy="false">(</mo><msup><mi>T</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^{0, \bullet}(T^{\otimes n})</annotation></semantics></math>) with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi mathvariant="normal">Hom</mi><mo stretchy="false">(</mo><msup><mi>T</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> R_n \;\in\; \Omega^{0, 1}\big( \mathrm{Hom}(T^{\otimes n}, T) \big) \,, </annotation></semantics></math></div> <p>make the shifted Dolbeault complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^{0, \bullet - 1}(T)</annotation></semantics></math> into an <a class="existingWikiWord" href="/nlab/show/L-infinity-algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebra</a> (called a “weak Lie algebra” in <a href="#Kapranov99">Kapranov 1999</a>).</p> <p></p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a>, <a class="existingWikiWord" href="/nlab/show/quadratic+form">quadratic form</a>, <a class="existingWikiWord" href="/nlab/show/sesquilinear+form">sesquilinear form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/Hermitian+form">Hermitian form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyper-K%C3%A4hler+manifold">hyper-Kähler manifold</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+K%C3%A4hler+manifold">quaternionic Kähler manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+potential">Kähler potential</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler-Einstein+manifold">Kähler-Einstein manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lefschetz+decomposition">Lefschetz decomposition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+polarization">Kähler polarization</a>, <a class="existingWikiWord" href="/nlab/show/almost+K%C3%A4hler+geometric+quantization">almost Kähler geometric quantization</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fedosov+deformation+quantization">Fedosov deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+spectral+sequence">Frölicher spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+manifold">Hodge manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sasakian+manifold">Sasakian manifold</a></p> </li> </ul> <div> <p><strong>classification of <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a> <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> by <a class="existingWikiWord" href="/nlab/show/Berger%27s+theorem">Berger's theorem</a>:</strong></p> <table><thead><tr><th></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math>preserved <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{C}\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/unitary+group">U(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>2</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,2n\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+forms">Kähler forms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\omega_2\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>2</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,2n\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℍ</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{H}\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/quaternionic+K%C3%A4hler+manifold">quaternionic Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Sp%28n%29.Sp%281%29">Sp(n).Sp(1)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>4</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,4n\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><msub><mi>ω</mi> <mn>4</mn></msub><mo>=</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>3</mn></msub><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/hyper-K%C3%A4hler+manifold">hyper-Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/symplectic+group">Sp(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>4</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,4n\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ω</mi><mo>=</mo><mi>a</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>b</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>c</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>b</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a^2 + b^2 + c^2 = 1</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>𝕆</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{O}\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29+manifold">Spin(7) manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math>8<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Cayley+form">Cayley form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82+manifold">G₂ manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mn>7</mn><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,7\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Kähler manifolds were first introduced and studied by P. A. Shirokov (cf. <a href="http://dx.doi.org/10.1070/RM1995v050n02ABEH002098">a historical article</a>) and later independently by Kähler.</p> <p>Textbook accounts:</p> <ul> <li id="Voisin02"> <p><a class="existingWikiWord" href="/nlab/show/Claire+Voisin">Claire Voisin</a>, section 3 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> <p><a class="existingWikiWord" href="/nlab/show/Ana+Cannas+da+Silva">Ana Cannas da Silva</a>, §16 in: <em>Lectures on Symplectic Geometry</em>, Lecture Notes in Mathematics <strong>1764</strong>, Springer (2008) [<a href="https://doi.org/10.1007/978-3-540-45330-7">doi:10.1007/978-3-540-45330-7</a>]</p> </li> </ul> <p>Lecture notes include</p> <ul> <li id="Moroianu07"> <p><a class="existingWikiWord" href="/nlab/show/Andrei+Moroianu">Andrei Moroianu</a>, <em>Lectures on Kähler Geometry</em>, Cambridge University Press 2007 (<a href="https://arxiv.org/abs/math/0402223">arXiv:math/0402223</a> <a href="https://doi.org/10.1017/CBO9780511618666">doi:10.1017/CBO9780511618666</a>, <a href="https://moroianu.perso.math.cnrs.fr/tex/kg.pdf">pdf</a>)</p> </li> <li id="Boalch09"> <p>Philip Boalch, <em>Noncompact complex symplectic and hyperkähler manifolds</em> (2009) [<a href="https://webusers.imj-prg.fr/~philip.boalch/cours09/hk.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Boalch-NoncompactSymplectic.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>Discussion in terms of <a class="existingWikiWord" href="/nlab/show/integrability+of+G-structure">first-order integrable</a> <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a> include</p> <ul> <li> <p><a href="#Moroianu07">Moroianu 07, 11.1</a></p> </li> <li id="Verbitsky09"> <p>Misha Verbitsky, <em>Kähler manifolds</em>, lecture notes 2009 (<a href="http://verbit.ru/MATH/TALKS/Unicamp-kahler-1.pdf">pdf</a>)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/spin+structures">spin structures</a> in Kähler manifolds is for instance in</p> <ul> <li id="Friedrich97"><a class="existingWikiWord" href="/nlab/show/Thomas+Friedrich">Thomas Friedrich</a>, <em>Dirac operators in Riemannian geometry</em>, Graduate studies in mathematics 25, AMS (1997)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a> on Kähler manifolds is in</p> <ul> <li id="BiquerdHoering08">O. Biquard, A. Höring, <em>Kähler geometry and Hodge theory</em>, 2008 (<a href="http://math.unice.fr/~hoering/hodge/hodge.pdf">pdf</a>)</li> </ul> <p>On Kapranov’s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_{\infty}</annotation></semantics></math>-structure:</p> <ul> <li id="Kapranov99"><a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, <em>Rozansky-Witten invariants via Atiyah classes</em>, Compositio Math. <strong>115</strong> 1 (1999) 71-113 [<a href="http://www.ams.org/mathscinet-getitem?mr=2000h:57056">MR2000h:57056</a>, <a href="http://dx.doi.org/10.1023/A:1000664527238">doi:10.1023/A:1000664527238</a>, <a href="http://arxiv.org/abs/alg-geom/9704009">arXiv:alg-geom/9704009</a>]</li> </ul> <p>For further developing of this concept in view of the <a class="existingWikiWord" href="/nlab/show/Fedosov+deformation+quantization">Fedosov deformation quantization</a>:</p> <ul> <li id="KwokwaiNaichungQin2020">Kwokwai Chan, Naichung Conan Leung, Qin Li, <em>Kapranov’s L∞ structures, Fedosov’s star products, and one-loop exact BV quantizations on Kähler manifolds</em>, <a href="https://arxiv.org/abs/2008.07057">abs/2008.07057</a></li> </ul> <p>On quantization of Kähler manifolds:</p> <ul> <li id="KwokwaiNaichungQin2020quant">Kwokwai Chan, Naichung Conan Leung, Qin Li, <em>Quantization of Kähler manifolds</em>, <a href="https://arxiv.org/abs/2009.03690">abs/2009.03690</a></li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+orbifolds">Kähler orbifolds</a>:</p> <ul> <li> <p>Thalia D. Jeffres, <em>Singular Set of Some Kähler Orbifolds</em>, Transactions of the American Mathematical Society Vol. 349, No. 5 (May, 1997), pp. 1961-1971 (<a href="https://www.jstor.org/stable/2155355">jstor:2155355</a>)</p> </li> <li> <p>Akira Fujiki, <em>On primitively symplectic compact Kähler V-manifolds</em>, in: <a class="existingWikiWord" href="/nlab/show/Kenji+Ueno">Kenji Ueno</a> (ed.), <em>Classification of Algebraic and Analytic Manifolds: Katata Symposium Proceedings 1982</em>, Birkhäuser 1983 (ISBN:9780817631376)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dominic+Joyce">Dominic Joyce</a>, Section 6.5.1 of: <em>Compact Manifolds with Special Holonomy</em>, Oxford Mathematical Monographs, Oxford University Press (2000) (<a href="https://global.oup.com/academic/product/compact-manifolds-with-special-holonomy-9780198506010?cc=de&lang=en&">ISBN:9780198506010</a>)</p> </li> <li> <p>Miguel Abreu, <em>Kähler Metrics on Toric Orbifolds</em>, J. Differential Geom. Volume 58, Number 1 (2001), 151-187 (<a href="https://projecteuclid.org/euclid.jdg/1090348285">euclid:jdg/1090348285</a>)</p> </li> <li> <p>Giovanni Bazzoni, Indranil Biswas, Marisa Fernández, Vicente Muñoz, Aleksy Tralle, <em>Homotopic properties of Kähler orbifolds</em>, In: Chiossi S., Fino A., Musso E., Podestà F., Vezzoni L. (eds.) <em>Special Metrics and Group Actions in Geometry</em> Springer INdAM Series, vol 23. Springer (2017) (<a href="https://arxiv.org/abs/1605.03024">arXiv:1605.03024</a>, <a href="https://doi.org/10.1007/978-3-319-67519-0_2">doi:10.1007/978-3-319-67519-0_2</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 24, 2024 at 10:34:13. 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