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<title>Lefschetz-Paket</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Lefschetz-Paket</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>In der <a href="Mathematik" title="Mathematik">Mathematik</a> ist das <b>Lefschetz-Paket</b> (auch <b>Hodge-Lefschetz-Paket</b> oder <b>Kähler-Paket</b>) ein abstraktes Prinzip, das sich in völlig unterschiedlichen Gebieten der Mathematik anwenden lässt und in jedem dieser Gebiete die Beweise tiefliegender Vermutungen ermöglicht.

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<div class="mw-heading mw-heading2"><h2 id="Abstrakte_Definition_eines_Lefschetz-Pakets">Abstrakte Definition eines Lefschetz-Pakets</h2></div>

<p>Zu einem mathematischen Objekt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> wird eine <a href="Graduierte_Algebra" class="mw-redirect" title="Graduierte Algebra">graduierte Algebra</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle A^{*}(X)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ffee42f5133d648735bf2f1997859abfb7ca69c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.587ex; height:2.843ex;" alt="{\displaystyle A^{*}(X)}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{0}(X)=\mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle A^{0}(X)=\mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca4598f4f7fe6f11fa94eaf8e52480e245ee1f40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.363ex; height:3.176ex;" alt="{\displaystyle A^{0}(X)=\mathbb {R} }" loading="lazy"></span> zugeordnet, so dass man für eine <a href="Ganze_Zahl" title="Ganze Zahl">ganze Zahl</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> (die „Dimension“ des Objekts <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>) und jede ganze Zahl <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> folgendes hat:

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<ul><li><a href="Poincar%C3%A9-Dualit%C3%A4t" title="Poincaré-Dualität">Poincaré-Dualität</a>: einen <a href="Vektorraum-Isomorphismus" class="mw-redirect" title="Vektorraum-Isomorphismus">Vektorraum-Isomorphismus</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle PD\colon A^{k}(X)\to A^{d-k}(X)^{*}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f552caaaf4cf65f751995ec34becc027c943c13f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.753ex; height:3.176ex;" alt="{\displaystyle PD\colon A^{k}(X)\to A^{d-k}(X)^{*}}" loading="lazy"></span>,</dd></dl></dd>

<dd>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44e23745a51c2c2d8d91fd98c1cf721573747ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle A^{*}}" loading="lazy"></span> den zu <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> <a href="Dualer_Vektorraum" class="mw-redirect" title="Dualer Vektorraum">dualen Vektorraum</a> bezeichnet.</dd></dl>

<ul><li><a href="Schwerer_Lefschetz-Satz" title="Schwerer Lefschetz-Satz">Schwerer Lefschetz-Satz</a>: eine lineare Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\colon A^{k}(X)\to A^{k+1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle L\colon A^{k}(X)\to A^{k+1}(X)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e69246efcd0003938cacf645e6e385b49d075681.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.574ex; height:3.176ex;" alt="{\displaystyle L\colon A^{k}(X)\to A^{k+1}(X)}" loading="lazy"></span>, so dass</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle HL:=L^{d-2k}\colon A^{k}(X)\to A^{d-k}(X)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/811b3c2353e3de55104b50ef3c653097cf82f871.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.052ex; height:3.176ex;" alt="{\displaystyle HL:=L^{d-2k}\colon A^{k}(X)\to A^{d-k}(X)}" loading="lazy"></span></dd></dl></dd>

<dd>ein Vektorraum-Isomorphismus ist.</dd></dl>

<ul><li>Hodge-Riemann-Relationen: die durch PD und HL gegebenen <a href="Nicht-ausgeartete_Bilinearform" class="mw-redirect" title="Nicht-ausgeartete Bilinearform">Paarung</a></li></ul>

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<annotation encoding="application/x-tex">{\displaystyle A^{k}(X)\times A^{k}(X)\to A^{d}(X)=\mathbb {R} }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0605985800f6d1385de598f07e69231f71f7c995.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.098ex; height:3.176ex;" alt="{\displaystyle A^{k}(X)\times A^{k}(X)\to A^{d}(X)=\mathbb {R} }" loading="lazy"></span></dd></dl></dd>

<dd>ist <a href="Symmetrische_Bilinearform" class="mw-redirect" title="Symmetrische Bilinearform">symmetrisch</a> sowie <a href="Positiv_definit" class="mw-redirect" title="Positiv definit">positiv definit</a> auf dem <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{d-2k+1}}">

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<annotation encoding="application/x-tex">{\displaystyle L^{d-2k+1}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5faf5e002a27e6ff65c1d44a1919cab052aad935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.732ex; height:2.676ex;" alt="{\displaystyle L^{d-2k+1}}" loading="lazy"></span>.</dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Kähler-Geometrie"><span id="K.C3.A4hler-Geometrie"></span>Kähler-Geometrie</h3></div>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> eine <a href="Geschlossene_Mannigfaltigkeit" title="Geschlossene Mannigfaltigkeit">geschlossene</a> <a href="K%C3%A4hler-Mannigfaltigkeit" title="Kähler-Mannigfaltigkeit">Kähler-Mannigfaltigkeit</a> mit <a href="K%C3%A4hlerform" class="mw-redirect" title="Kählerform">Kählerform</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>. Dann wird durch die klassische <a href="Poincar%C3%A9-Dualit%C3%A4t" title="Poincaré-Dualität">Poincaré-Dualität</a> und die durch

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<annotation encoding="application/x-tex">{\displaystyle L(.):=\omega \wedge .}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f291f72898b4d08ec1f0860705f2e11089203f28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.814ex; height:2.843ex;" alt="{\displaystyle L(.):=\omega \wedge .}" loading="lazy"></span></dd></dl>

<p>induzierte lineare Abbildung <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> ein Lefschetz-Paket auf der <a href="De-Rham-Kohomologie" title="De-Rham-Kohomologie">De-Rham-Kohomologie</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{dR}^{*}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle H_{dR}^{*}(M)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5edd9b39da4e880340f9e55e394bf76d3dea4929.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.522ex; height:3.009ex;" alt="{\displaystyle H_{dR}^{*}(M)}" loading="lazy"></span> definiert. Dies hat zahlreiche Anwendungen in der Theorie der Kähler-Mannigfaltigkeiten, unter anderem den <a href="Indexsatz_von_Hodge" title="Indexsatz von Hodge">Hodge-Indexsatz</a> und die Konstruktion und Eigenschaften von Periodenabbildungen.

</p><p>Weiterhin definiert die Einschränkung von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> auch ein Lefschetz-Paket auf der <a href="Dolbeault-Kohomologie" title="Dolbeault-Kohomologie">Dolbeault-Kohomologie</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigoplus _{k}H^{k,k}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \bigoplus _{k}H^{k,k}(M)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f07ad0ae4248ead28814967775a7066207b3d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.815ex; height:3.176ex;" alt="{\displaystyle \textstyle \bigoplus _{k}H^{k,k}(M)}" loading="lazy"></span>.

Allgemeiner kann man zu komplexen<a href="Projektive_Variet%C3%A4t" title="Projektive Varietät"> projektiven Varietäten</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> ihre Schnittkohomologie betrachten und erhält dann ebenfalls eine Lefschetz-Zerlegung auf <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{k}H^{k,k}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{k}H^{k,k}(X)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b917e7eac36e6612635150a0642e2c808379c5ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.193ex; height:5.509ex;" alt="{\displaystyle \bigoplus _{k}H^{k,k}(X)}" loading="lazy"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Algebraische_Geometrie">Algebraische Geometrie</h3></div>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> eine <a href="Algebraische_Variet%C3%A4t" title="Algebraische Varietät">algebraische Varietät</a>. Grothendiecks Standardvermutungen besagen, dass man ein Lefschetz-Paket auf dem Vektorraum der algebraischen Zykel moduli homologischer Äquivalenz hat. Sie sind unbewiesen. Aus den Standardvermutungen folgen die von <a href="Pierre_Deligne" title="Pierre Deligne">Deligne</a> mit anderen Methoden bewiesenen <a href="Weil-Vermutung" title="Weil-Vermutung">Weil-Vermutungen</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Polytope_und_triangulierte_Sphären"><span id="Polytope_und_triangulierte_Sph.C3.A4ren"></span>Polytope und triangulierte Sphären</h3></div>

<p>Die kombinatorische Schnittkohomologie eines <a href="Konvexe_Menge" title="Konvexe Menge">konvexen</a> <a href="Polytop_(Geometrie)" title="Polytop (Geometrie)">Polytops</a> hat ein Lefschetz-Paket. Mit dem schweren Lefschetz-Satz bewies <a href="Richard_P._Stanley" title="Richard P. Stanley">Stanley</a> die g-Vermutung für simpliziale Polytope<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. Kalle Karu erweiterte dies auf allgemeine Polytope<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>, und <a href="Karim_Adiprasito" title="Karim Adiprasito">Adiprasito</a> zeigte die g-Vermutung für triangulierte Sphären<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>. Aus den Hodge-Riemann-Relationen folgt die Alexandrov-Fenchel-Ungleichung.

</p>

<div class="mw-heading mw-heading3"><h3 id="Darstellungstheorie">Darstellungstheorie</h3></div>

<p>Soergelsche Bimoduln haben ein Lefschetz-Paket. Daraus folgt die Positivität der Koeffizienten der <a href="Kazhdan-Lusztig-Polynom" title="Kazhdan-Lusztig-Polynom">Kazhdan-Lusztig-Polynome</a> sowie ein algebraischer Beweis (<a href="Geordie_Williamson" title="Geordie Williamson">Geordie Williamson</a>, <a href="Ben_Elias_(Mathematiker)" title="Ben Elias (Mathematiker)">Ben Elias</a>) der zuvor von <a href="Alexander_Beilinson" title="Alexander Beilinson">Beilinson</a>-<a href="Joseph_Bernstein" title="Joseph Bernstein">Bernstein</a>, <a href="Jean-Luc_Brylinski" title="Jean-Luc Brylinski">Brylinski</a>-<a href="Masaki_Kashiwara" title="Masaki Kashiwara">Kashiwara</a> und später <a href="Wolfgang_Soergel_(Mathematiker)" title="Wolfgang Soergel (Mathematiker)">Soergel</a> mit anderen Methoden bewiesenen Kazhdan-Lusztig-Vermutung, einer Charakterformel für <a href="Satz_vom_h%C3%B6chsten_Gewicht" title="Satz vom höchsten Gewicht">Darstellungen höchsten Gewichts</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Matroide">Matroide</h3></div>

<p>Der <a href="Chow-Ring" class="mw-redirect" title="Chow-Ring">Chow-Ring</a> eines <a href="Matroid" title="Matroid">Matroids</a> hat ein Lefschetz-Paket. Aus den Hodge-Riemann-Relationen folgt, dass die Folge der Koeffizienten des chromatischen Polynoms des Matroids <a href="Log-konkave_Folge" class="mw-redirect" title="Log-konkave Folge">log-konkav</a> und damit <a href="Unimodale_Folge" title="Unimodale Folge">unimodal</a> ist.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Für eine Folge reeller Zahlen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bc77764b2e74e64a63341054fa90f3e07db275f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}" loading="lazy"></span> bedeutet log-konkav, dass für die Folgenglieder <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i+1}a_{i-1}\leq a_{i}^{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{i+1}a_{i-1}\leq a_{i}^{2}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/590abd16d02bb776bd38eced7d4274619ffadfa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.643ex; height:3.176ex;" alt="{\displaystyle a_{i+1}a_{i-1}\leq a_{i}^{2}}" loading="lazy"></span> gilt, und unimodular, dass es ein Folgenglied <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05e256a120c3ab9f8958de71acdf81cd75065e3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}" loading="lazy"></span> gibt, so dass (die Folge bestehe aus n Folgengliedern) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}\leq \cdots \leq a_{k-1}\leq a_{k}\geq a_{k+1}\geq \cdots \geq a_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1}\leq \cdots \leq a_{k-1}\leq a_{k}\geq a_{k+1}\geq \cdots \geq a_{n}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a39878d9815c54ad6823b9c941a4a04bb124845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.926ex; height:2.343ex;" alt="{\displaystyle a_{1}\leq \cdots \leq a_{k-1}\leq a_{k}\geq a_{k+1}\geq \cdots \geq a_{n}}" loading="lazy"></span>, das heißt sie hat ein Maximum und ist ansonsten auf der einen Seite monoton fallend und auf der anderen monoton steigend. Diese Eigenschaften gelten insbesondere für die Koeffizienten des <a href="Chromatisches_Polynom" title="Chromatisches Polynom">chromatischen Polynoms</a> von <a href="Graph_(Graphentheorie)" title="Graph (Graphentheorie)">Graphen</a>, eine Vermutung von <a href="Ronald_C._Read" title="Ronald C. Read">Ronald C. Read</a>, die <a href="June_Huh" title="June Huh">June Huh</a> vor dem Beweis des allgemeineren Falls der Matroide bewiesen hat.

</p>

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

<ul><li><a href="Claire_Voisin" title="Claire Voisin">Claire Voisin</a>: Hodge theory and the topology of compact Kähler and complex projective manifolds. <a rel="nofollow" class="external text" href="https://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf">online</a></li>

<li>June Huh: Tropical geometry of matroids. <a rel="nofollow" class="external text" href="https://web.math.princeton.edu/~huh/TropicalMatroids.pdf">online</a></li>

<li>June Huh: Combinatorial applications of the Hodge-Riemann relations, Proc. ICM 2018, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1711.11176">Arxiv</a></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">A. Beilinson, J. Bernstein, P. Deligne: <i>Faisceaux pervers</i>, Asterisque (1982)</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Kleiman: <i>Algebraic cycles and the Weil conjectures</i>. (<a rel="nofollow" class="external text" href="https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Fourothers.pdf">online</a>)</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">R. Stanley: <i>The number of faces of a simplicial convex polytope</i>, Adv. Math. 35, 236-238 (1980)</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">K. Karu: <i>Hard Lefschetz theorem for nonrational polytopes</i>, Invent. Math. 157, 419-447 (2004)</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><span class="cite">Gil Kalai: <a rel="nofollow" class="external text" href="https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/"><i>Amazing: Karim Adiprasito proved the g-conjecture for spheres!</i></a> In: <i>Combinatorics and more.</i> 25. Dezember 2018,<span class="Abrufdatum"> abgerufen am 26. Januar 2019</span> (englisch).</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ALefschetz-Paket&rft.title=Amazing%3A+Karim+Adiprasito+proved+the+g-conjecture+for+spheres%21&rft.description=Amazing%3A+Karim+Adiprasito+proved+the+g-conjecture+for+spheres%21&rft.identifier=https%3A%2F%2Fgilkalai.wordpress.com%2F2018%2F12%2F25%2Famazing-karim-adiprasito-proved-the-g-conjecture-for-spheres%2F&rft.creator=Gil+Kalai&rft.date=2018-12-25&rft.language=en"> </span></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">B. Elias, G. Williamson: <i>The Hodge theory of Soergel bimodules</i>, Ann. Math. 180, 1089-1136 (2014)</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">K. Adiprasito, J. Huh, E. Katz: <i>Hodge theory for combinatorial geometries</i>, Ann. Math. 188 (2018)</span>

</li>

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