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<title>Komplexe Differentialform</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Komplexe Differentialform</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">Eine komplexe Differentialform ist ein <a href="Mathematisches_Objekt" title="Mathematisches Objekt">mathematisches Objekt</a> aus der <a href="Komplexe_Geometrie" class="mw-redirect" title="Komplexe Geometrie">komplexen Geometrie</a>. Eine komplexe Differentialform ist eine Entsprechung der <a href="Differentialform" title="Differentialform">(reellen) Differentialformen</a> auf <a href="Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit">komplexen Mannigfaltigkeiten</a>. Genauso wie im reellen Fall bilden auch die komplexen Differentialform eine <a href="Graduierte_Algebra" class="mw-redirect" title="Graduierte Algebra">graduierte Algebra</a>. Eine komplexe Differentialform vom Grad <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><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"> (oder kurz k-Form) kann auf eindeutige Art und Weise in zwei Differentialformen zerlegt werden, die dann den Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> beziehungsweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q=k}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/890688054abf6d2d38c37c03619b5b3436577dd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.478ex; height:2.509ex;" alt="{\displaystyle p+q=k}" loading="lazy"> haben. Um diese Zerlegung zu betonen, spricht man auch von (p,q)-Formen. Bei dieser kurzen Sprechweise wird auch klar, dass es sich um komplexe Differentialformen handelt, denn reelle Formen besitzen keine solche Zerlegung. Eine wichtige Rolle spielt der <a href="Kalk%C3%BCl" title="Kalkül">Kalkül</a> der komplexen Differentialformen in der <a href="Hodge-Theorie" class="mw-redirect" title="Hodge-Theorie">Hodge-Theorie</a>.

<div class="mw-heading mw-heading2"><h2 id="Komplexe_Differentialformen">Komplexe Differentialformen</h2></div>

Sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><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"> eine <a href="Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit">komplexe Mannigfaltigkeit</a> der (komplexen) Dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. Wähle

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<annotation encoding="application/x-tex">{\displaystyle \{\mathrm {d} z^{j}=dx^{j}+idy^{j},\ \mathrm {d} {\bar {z}}^{j}=dx^{j}-idy^{j};\ 1\leq j\leq n\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fff0effaf9cdf0765f376d91cf441d519f8e428.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.023ex; height:3.176ex;" alt="{\displaystyle \{\mathrm {d} z^{j}=dx^{j}+idy^{j},\ \mathrm {d} {\bar {z}}^{j}=dx^{j}-idy^{j};\ 1\leq j\leq n\}}" loading="lazy"></dd></dl>

als eine lokale Basis des komplexifizierten <a href="Kotangentialraum" title="Kotangentialraum">Kotangentialraums</a>. Die <a href="Kovektor" class="mw-redirect" title="Kovektor">Kovektoren</a> haben die lokale Darstellung

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<annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{n}f_{j}\mathrm {d} z^{j}+g_{j}\mathrm {d} {\overline {z}}^{j}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ca7140b8058bb29b485fff9a4727f89d3f395ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18ex; height:7.176ex;" alt="{\displaystyle \sum _{j=1}^{n}f_{j}\mathrm {d} z^{j}+g_{j}\mathrm {d} {\overline {z}}^{j}.}" loading="lazy"></dd></dl>

Die Räume, in denen nur Basisvektoren der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathrm {d} z^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathrm {d} z^{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3f8c6df2cb9e97d3d871cf783a089d8766014f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.293ex; height:2.509ex;" alt="{\displaystyle \textstyle \mathrm {d} z^{j}}" loading="lazy"> vorkommen, werden verbal als (1,0)-Formen und formelmäßig mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}^{1,0}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}^{1,0}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1175bb20ab3f9f90bee3d6e11ea6b85f9ea5586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:3.176ex;" alt="{\displaystyle {\mathcal {A}}^{1,0}(M)}" loading="lazy"> bezeichnet. Analog dazu ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}^{0,1}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}^{0,1}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e17933eae37d235e57bfd31803077ecd90c3a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:3.176ex;" alt="{\displaystyle {\mathcal {A}}^{0,1}(M)}" loading="lazy"> der Raum der (0,1)-Formen, also der Kovektoren, welche nur Basisvektoren der Form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathrm {d} {\overline {z}}^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathrm {d} {\overline {z}}^{j}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23f184275aaead92c27dad3f47a7e265b3501489.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.41ex; height:2.843ex;" alt="{\displaystyle \textstyle \mathrm {d} {\overline {z}}^{j}}" loading="lazy"> haben. Diese beiden Räume sind stabil, das heißt unter holomorphen Koordinatenwechseln werden diese Räume in sich selbst abgebildet. Aus diesem Grund sind die Räume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}^{1,0}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}^{1,0}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1175bb20ab3f9f90bee3d6e11ea6b85f9ea5586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:3.176ex;" alt="{\displaystyle {\mathcal {A}}^{1,0}(M)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}^{0,1}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}^{0,1}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e17933eae37d235e57bfd31803077ecd90c3a16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.503ex; height:3.176ex;" alt="{\displaystyle {\mathcal {A}}^{0,1}(M)}" loading="lazy"> komplexe <a href="Vektorb%C3%BCndel" title="Vektorbündel">Vektorbündel</a> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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Mit Hilfe des <a href="%C3%84u%C3%9Fere_Algebra" class="mw-redirect" title="Äußere Algebra">äußeren Produktes</a> von komplexen Differentialformen, welches genauso wie für reelle Differentialformen definiert ist, kann man nun die Räume der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)}">

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<annotation encoding="application/x-tex">{\displaystyle (p,q)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}^{p,q}(M):=\bigwedge _{j=1}^{p}{\mathcal {A}}^{1,0}(M)\wedge \bigwedge _{j=1}^{q}{\mathcal {A}}^{0,1}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c51b57b93682e061231fc948e451d42252eb24bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.107ex; height:7.343ex;" alt="{\displaystyle {\mathcal {A}}^{p,q}(M):=\bigwedge _{j=1}^{p}{\mathcal {A}}^{1,0}(M)\wedge \bigwedge _{j=1}^{q}{\mathcal {A}}^{0,1}(M)}" loading="lazy"></dd></dl>

definieren. Weiter definiert man noch den Raum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}^{r}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}^{r}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f42dce926a619b9ce15650a3d9aeaefa1100a676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.562ex; height:2.843ex;" alt="{\displaystyle {\mathcal {E}}^{r}(M)}" loading="lazy"> als die <a href="Direkte_Summe" title="Direkte Summe">direkte Summe</a>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}^{r}(M):=\bigoplus _{p+q=r}{\mathcal {A}}^{p,q}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a52f96ef4292d98ace1301b25e1d259d8ee6c52b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.018ex; height:5.676ex;" alt="{\displaystyle {\mathcal {E}}^{r}(M):=\bigoplus _{p+q=r}{\mathcal {A}}^{p,q}(M)}" loading="lazy"></dd></dl>

der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)}">

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<annotation encoding="application/x-tex">{\displaystyle (p,q)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9769c58523b9b639866a2d48e657d9c26911143a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (p,q)}" loading="lazy">-Formen mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=p+q}">

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<annotation encoding="application/x-tex">{\displaystyle r=p+q}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db5a84278bff4da59f96a7c9a0a5d0c28366e099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.226ex; height:2.343ex;" alt="{\displaystyle r=p+q}" loading="lazy">. Dies ist isomorph zur direkten Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}^{r}(M)\cong {\mathcal {A}}^{r}(M)\oplus i{\mathcal {A}}^{r}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}^{r}(M)\cong {\mathcal {A}}^{r}(M)\oplus i{\mathcal {A}}^{r}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b60b3594d92411c54776d1b3fd7fa4af707cca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.59ex; height:2.843ex;" alt="{\displaystyle {\mathcal {E}}^{r}(M)\cong {\mathcal {A}}^{r}(M)\oplus i{\mathcal {A}}^{r}(M)}" loading="lazy"> der Räume der reellen Differentialformen. Außerdem ist für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q=r}">

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<annotation encoding="application/x-tex">{\displaystyle p+q=r}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d77def8669646a28418f13507614a3e8a2839d2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.316ex; height:2.343ex;" alt="{\displaystyle p+q=r}" loading="lazy"> eine Projektion

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<annotation encoding="application/x-tex">{\displaystyle \pi _{p,q}\colon {\mathcal {E}}^{r}(M)\to {\mathcal {A}}^{p,q}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0396600477b7c1067d11c94b267e371073cdaada.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.25ex; height:3.009ex;" alt="{\displaystyle \pi _{p,q}\colon {\mathcal {E}}^{r}(M)\to {\mathcal {A}}^{p,q}(M)}" loading="lazy"></dd></dl>

definiert, welche jeder komplexen Differentialform vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> ihre <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)}">

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Eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9769c58523b9b639866a2d48e657d9c26911143a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (p,q)}" loading="lazy">-Form hat also in lokalen Koordinaten <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{1},\ldots ,z_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle z_{1},\ldots ,z_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/399be1c816346daeabf844df4597179a1c8bbdf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.613ex; height:2.009ex;" alt="{\displaystyle z_{1},\ldots ,z_{n}}" loading="lazy"> die eindeutige Darstellung

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<annotation encoding="application/x-tex">{\displaystyle \omega =\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/57ea265bab154e0cc459003f65e6f3f7d27f7d47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:64.325ex; height:8.009ex;" alt="{\displaystyle \omega =\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}.}" loading="lazy"></dd></dl>

Da diese Darstellung doch sehr lang ist, ist es üblich die Kurzschreibweise

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<annotation encoding="application/x-tex">{\displaystyle \sum _{I,J}f_{I,J}\mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}:=\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}^{}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd612fc5e1b58eeadfdf64fadc9654aa39f9dbe7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:80.11ex; height:8.676ex;" alt="{\displaystyle \sum _{I,J}f_{I,J}\mathrm {d} z_{I}\wedge \mathrm {d} {\overline {z}}_{J}:=\sum _{\stackrel {1\leq i_{1}<\ldots <i_{p}\leq p}{1\leq j_{1}<\ldots <j_{q}\leq q}}^{}f_{i_{1},\ldots i_{p},j_{1},\ldots j_{q}}\mathrm {d} z_{i_{1}}\wedge \cdots \wedge \mathrm {d} z_{i_{p}}\wedge \mathrm {d} {\overline {z}}_{j_{1}}\wedge \cdots \wedge \mathrm {d} {\overline {z}}_{j_{q}}}" loading="lazy"></dd></dl>

zu vereinbaren.

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

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

Die äußere Ableitung

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} :{\mathcal {E}}^{r}(M)\to {\mathcal {E}}^{r+1}(M),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e03c08266d1cecbeacddc7fb4e08842875ed300e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.714ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} :{\mathcal {E}}^{r}(M)\to {\mathcal {E}}^{r+1}(M),}" loading="lazy"></dd></dl>

was gleichbedeutend ist mit

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} :\bigoplus _{p+q=r}{\mathcal {A}}^{p,q}(M)\to \bigoplus _{p+q=r}{\mathcal {A}}^{p+1,q}(M)\oplus \bigoplus _{p+q=r}{\mathcal {A}}^{p,q+1}(M),}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b98a447b68523463f433169d2b3468970586120.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.666ex; height:5.676ex;" alt="{\displaystyle \mathrm {d} :\bigoplus _{p+q=r}{\mathcal {A}}^{p,q}(M)\to \bigoplus _{p+q=r}{\mathcal {A}}^{p+1,q}(M)\oplus \bigoplus _{p+q=r}{\mathcal {A}}^{p,q+1}(M),}" loading="lazy"></dd></dl>

kann in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} =\partial +{\overline {\partial }}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/762d2ab184755313884f6f70352578528d2ac993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.037ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} =\partial +{\overline {\partial }}}" loading="lazy"> aufgespalten werden. Die Dolbeault-Operatoren

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<annotation encoding="application/x-tex">{\displaystyle \partial :{\mathcal {A}}^{p,q}(M)\to {\mathcal {A}}^{p+1,q}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87c160f498ba540abbb84f0b96bd074f3dc072a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.855ex; height:3.176ex;" alt="{\displaystyle \partial :{\mathcal {A}}^{p,q}(M)\to {\mathcal {A}}^{p+1,q}(M)}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}:{\mathcal {A}}^{p,q}(M)\to {\mathcal {A}}^{p,q+1}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e832de2700b8d3be3d38001389696541b97ed30b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.024ex; height:3.509ex;" alt="{\displaystyle {\overline {\partial }}:{\mathcal {A}}^{p,q}(M)\to {\mathcal {A}}^{p,q+1}(M)}" loading="lazy"></dd></dl>

sind definiert durch

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\partial &:=\pi _{p+1,q}\circ \mathrm {d} \\{\overline {\partial }}&:=\pi _{p,q+1}\circ \mathrm {d} .\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b80e0f04f1c461c6efa5a2630a79d272a1c13a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.586ex; margin-bottom: -0.252ex; width:15.817ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\partial &:=\pi _{p+1,q}\circ \mathrm {d} \\{\overline {\partial }}&:=\pi _{p,q+1}\circ \mathrm {d} .\end{aligned}}}" loading="lazy"></dd></dl>

In lokalen Koordinaten bedeutet dies

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<annotation encoding="application/x-tex">{\displaystyle \partial \left(\sum _{I,J}f_{I,J}\mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}\right)=\sum _{I,J}\sum _{l=1}^{n}{\frac {\partial f_{I,J}}{\partial z^{l}}}\mathrm {d} z^{l}\wedge \mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c1901fe8ae950df0f421c2b9201762e87890635.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:54.547ex; height:7.676ex;" alt="{\displaystyle \partial \left(\sum _{I,J}f_{I,J}\mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}\right)=\sum _{I,J}\sum _{l=1}^{n}{\frac {\partial f_{I,J}}{\partial z^{l}}}\mathrm {d} z^{l}\wedge \mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}\left(\sum _{I,J}f_{I,J}\mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}\right)=\sum _{I,J}\sum _{l=1}^{n}{\frac {\partial f_{I,J}}{\partial {\overline {z}}^{l}}}\mathrm {d} {\overline {z}}^{l}\wedge \mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/634c6c37268526bd005a27577bdaa2a372b85ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.481ex; height:7.676ex;" alt="{\displaystyle {\overline {\partial }}\left(\sum _{I,J}f_{I,J}\mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}\right)=\sum _{I,J}\sum _{l=1}^{n}{\frac {\partial f_{I,J}}{\partial {\overline {z}}^{l}}}\mathrm {d} {\overline {z}}^{l}\wedge \mathrm {d} z^{I}\wedge \mathrm {d} {\overline {z}}^{J}.}" loading="lazy"></dd></dl>

Dabei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/62b4e7c1cedb9564609aefd2aa2309972f455c24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:2.176ex;" alt="{\displaystyle \partial }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\partial }}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bd5dfe63f32552004b0d29d5ef7e8b046dda0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.487ex; height:3.009ex;" alt="{\displaystyle {\overline {\partial }}}" loading="lazy"> auf den rechten Seiten der Gleichung die normalen <a href="Dolbeault-Operator" class="mw-redirect" title="Dolbeault-Operator">Dolbeault-Operatoren</a>.

<div class="mw-heading mw-heading3"><h3 id="Holomorphe_Differentialformen">Holomorphe Differentialformen</h3></div>

Erfüllt eine Differentialform <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in {\mathcal {A}}^{p,0}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle \omega \in {\mathcal {A}}^{p,0}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ae5c05eb51ef2ec3e7db9c0063af144178f367d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.795ex; height:3.176ex;" alt="{\displaystyle \omega \in {\mathcal {A}}^{p,0}(M)}" loading="lazy"> die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\partial }}\omega =0}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}\omega =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a786bc92348ae105b9e8911702c28404313978c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.194ex; height:3.009ex;" alt="{\displaystyle {\overline {\partial }}\omega =0}" loading="lazy">, so spricht man von einer holomorphen Differentialform. In lokalen Koordinaten kann man diese Formen durch

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<annotation encoding="application/x-tex">{\displaystyle \sum _{I}f_{I}\mathrm {d} z^{I}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f421de848181d6b270237b783b02aee4480e81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.386ex; height:5.509ex;" alt="{\displaystyle \sum _{I}f_{I}\mathrm {d} z^{I}}" loading="lazy"></dd></dl>

darstellen, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{I}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/261d1a55930b021df6465cfd506dc13fc00b491d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.2ex; height:2.509ex;" alt="{\displaystyle f_{I}}" loading="lazy"> holomorphe Funktionen sind. Der <a href="Vektorraum" title="Vektorraum">Vektorraum</a> der holomorphen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-Formen auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><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"> wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{p}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega ^{p}(M)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4e1ca1779b05e792168050e8e356c923fe2731a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.989ex; height:2.843ex;" alt="{\displaystyle \Omega ^{p}(M)}" loading="lazy"> notiert.

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

<ul><li>Für diese Operatoren gilt eine <a href="Produktregel" title="Produktregel">Leibniz-Regel</a>. Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \in {\mathcal {A}}^{p,q}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega \in {\mathcal {A}}^{p,q}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7a76619af6fc36613eb506b691bf938049fcf13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.477ex; height:2.343ex;" alt="{\displaystyle \omega \in {\mathcal {A}}^{p,q}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \in {\mathcal {A}}^{r,s}}">

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<annotation encoding="application/x-tex">{\displaystyle \nu \in {\mathcal {A}}^{r,s}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add41606eee7ee3ccad3f1edf47c2ae7bc3c8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.193ex; height:2.343ex;" alt="{\displaystyle \nu \in {\mathcal {A}}^{r,s}}" loading="lazy">, dann gilt</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \partial (\omega \wedge \nu )=\partial \omega \wedge \nu +(-1)^{p+q}\,\omega \wedge \partial \nu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f823ef7aca63f6de4c030ef48e06daa862250642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.745ex; height:3.009ex;" alt="{\displaystyle \partial (\omega \wedge \nu )=\partial \omega \wedge \nu +(-1)^{p+q}\,\omega \wedge \partial \nu }" loading="lazy"></dd></dl></dd>

<dd>und

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}(\omega \wedge \nu )={\overline {\partial }}\omega \wedge \nu +(-1)^{p+q}\,\omega \wedge {\overline {\partial }}\nu .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e561bcca32077b8ae0be705b9bc24d63955690c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.9ex; height:3.509ex;" alt="{\displaystyle {\overline {\partial }}(\omega \wedge \nu )={\overline {\partial }}\omega \wedge \nu +(-1)^{p+q}\,\omega \wedge {\overline {\partial }}\nu .}" loading="lazy"></dd></dl></dd></dl>

<ul><li>Aus der Identität</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle 0=\mathrm {d} ^{2}=(\partial +{\overline {\partial }})^{2}=\partial ^{2}+({\overline {\partial }}\partial +\partial {\overline {\partial }})+{\overline {\partial }}^{2}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a99cbd2758e42dfc3c2f5944f8d6e71f7a2e442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.195ex; height:4.009ex;" alt="{\displaystyle 0=\mathrm {d} ^{2}=(\partial +{\overline {\partial }})^{2}=\partial ^{2}+({\overline {\partial }}\partial +\partial {\overline {\partial }})+{\overline {\partial }}^{2}}" loading="lazy"></dd></dl></dd>

<dd>folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial ^{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \partial ^{2}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e905f7d85950dffd9d39b510a0f25ff646de12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.658ex; height:2.676ex;" alt="{\displaystyle \partial ^{2}=0}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial {\overline {\partial }}+{\overline {\partial }}\partial =0}">

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<annotation encoding="application/x-tex">{\displaystyle \partial {\overline {\partial }}+{\overline {\partial }}\partial =0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e55245872c442cb57c1cf8ec7441132c0a72a28a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.712ex; height:3.176ex;" alt="{\displaystyle \partial {\overline {\partial }}+{\overline {\partial }}\partial =0}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\partial }}^{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle {\overline {\partial }}^{2}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf23d6689179d8e57396bb5a2c87f8ed2b91b942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.803ex; height:3.509ex;" alt="{\displaystyle {\overline {\partial }}^{2}=0}" loading="lazy">, denn alle drei Terme sind von unterschiedlichem Grad. Die Operatoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial }">

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bd5dfe63f32552004b0d29d5ef7e8b046dda0e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.487ex; height:3.009ex;" alt="{\displaystyle {\overline {\partial }}}" loading="lazy"> eignen sich also für eine Kohomologietheorie. Diese trägt den Namen <a href="Dolbeault-Kohomologie" title="Dolbeault-Kohomologie">Dolbeault-Kohomologie</a>.</dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68e27d2e539fd0c3a9a7efab6257abd17de7fc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.401ex; height:2.843ex;" alt="{\displaystyle (M,g)}" loading="lazy"> eine <a href="K%C3%A4hlermannigfaltigkeit" class="mw-redirect" title="Kählermannigfaltigkeit">Kählermannigfaltigkeit</a>, also eine <a href="Komplexe_Mannigfaltigkeit" title="Komplexe Mannigfaltigkeit">komplexe Mannigfaltigkeit</a> mit einer verträglichen Riemann’schen Metrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy">, so kann man den <a href="Adjungierter_Operator" title="Adjungierter Operator">adjungierten</a> Dolbeault-Quer-Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\partial }}^{*}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e534223641821b90cdcaaa2a0fcacd15806e348.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.542ex; height:3.176ex;" alt="{\displaystyle {\overline {\partial }}^{*}}" loading="lazy"> bezüglich dieser Metrik bilden. Der Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta :={\overline {\partial }}\,{\overline {\partial }}^{*}+{\overline {\partial }}^{*}{\overline {\partial }}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta :={\overline {\partial }}\,{\overline {\partial }}^{*}+{\overline {\partial }}^{*}{\overline {\partial }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbc99ff23c31531e120762deb31b9387d7989a1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.967ex; height:3.343ex;" alt="{\displaystyle \Delta :={\overline {\partial }}\,{\overline {\partial }}^{*}+{\overline {\partial }}^{*}{\overline {\partial }}}" loading="lazy"> ist dann ein <a href="Verallgemeinerter_Laplace-Operator" title="Verallgemeinerter Laplace-Operator">verallgemeinerter Laplace-Operator</a>. Anwendung findet dieser Operator in der (komplexen) <a href="Hodge-Theorie" class="mw-redirect" title="Hodge-Theorie">Hodge-Theorie</a>.</li></ul>

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

<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ComplexForm.html">Complex Form.</a> In: <a href="MathWorld" title="MathWorld">MathWorld</a> (englisch).</li>

<li>Raymond O. Wells: Differential analysis on complex manifolds. Prentice-Hall, Englewood Cliffs NJ 1973, ISBN 0-13-211508-5.</li>

<li><a href="Lars_H%C3%B6rmander" title="Lars Hörmander">Lars Hörmander</a>: An Introduction to Complex Analysis in Several Variables (= North-Holland Mathematical Library 7). 2. revised edition. North-Holland u. a., Amsterdam u. a. 1973, ISBN 0-7204-2450-X.</li></ul></div><div><div class="zim-footer">

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