• geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be...
    20 KB (4,546 words) - 05:19, 1 June 2023
  • Thumbnail for De Rham cohomology
    other lies the de Rham cohomology. The de Rham cohomology has inspired many mathematical ideas, including Dolbeault cohomology, Hodge theory, and the...
    19 KB (2,923 words) - 23:19, 2 May 2025
  • Thumbnail for Pierre Dolbeault
    an Analysis seminar in Paris. Dolbeault cohomology is named after him, and so is the Dolbeault theorem. Pierre Dolbeault's professional webpage "On the...
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  • algebraic varieties, by an application of Dolbeault's theorem relating sheaf cohomology to Dolbeault cohomology. Let X be a smooth variety of dimension...
    18 KB (3,295 words) - 01:37, 25 May 2025
  • form the basis for Dolbeault cohomology and many aspects of Hodge theory. On a star-shaped domain of a complex manifold the Dolbeault operators have dual...
    9 KB (1,413 words) - 02:38, 27 April 2024
  • }}} -lemma holds. In particular the Bott–Chern cohomology is an alternative to the Dolbeault cohomology of a compact complex manifolds, and they are isomorphic...
    33 KB (4,739 words) - 20:31, 30 April 2025
  • Bott–Chern cohomology to Dolbeault cohomology are isomorphisms, and furthermore the map from Bott–Chern cohomology to de Rham cohomology is injective...
    18 KB (2,991 words) - 08:15, 17 February 2024
  • sequence describes the precise relationship between the Dolbeault cohomology and the de Rham cohomology of a general complex manifold. On a compact Kähler...
    7 KB (1,027 words) - 11:27, 9 June 2025
  • counterexamples on contractible domains which have non-vanishing first Dolbeault cohomology, see the post https://mathoverflow.net/a/59554. Warner 1983, pp. 155–156...
    29 KB (5,411 words) - 11:17, 4 May 2025
  • }}^{1})} where we have applied the Dolbeault theorem to phrase the Dolbeault cohomology in terms of sheaf cohomology of the sheaf of holomorphic ( 1 ,...
    31 KB (5,131 words) - 02:41, 29 March 2025
  • of Hodge theory to produce a number of relations on de Rham and Dolbeault cohomology of compact Kähler manifolds, such as the Lefschetz hyperplane theorem...
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  • well as sheaf extension. By Dolbeault's theorem, this sheaf cohomology can alternatively be described as the cohomology of the chain complex defined...
    13 KB (2,394 words) - 19:12, 28 January 2025
  • Frölicher spectral sequence starting from the Dolbeault cohomology and converging to the algebraic de Rham cohomology of a variety. Hodge–de Rham spectral sequence...
    51 KB (10,712 words) - 03:03, 6 July 2025
  • ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}}^{\star })} and we can apply the Dolbeault isomorphism to calculate H 1 ( C n , O C n ) ≃ H 1 ( C n , Ω C n 0 ) ≃...
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  • ^{n}} by Takeuchi. Bicomplex number Complex geometry CR manifold Dolbeault cohomology Harmonic maps Harmonic morphisms Infinite-dimensional holomorphy...
    124 KB (17,717 words) - 22:01, 1 July 2025
  • Willmore energy Willmore flow Atiyah–Singer index theorem de Rham cohomology Dolbeault cohomology elliptic complex Hodge theory pseudodifferential operator Klein...
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  • theory and Dolbeault's theorem. Hodge–de Rham spectral sequence Frölicher, Alfred (1955), "Relations between the cohomology groups of Dolbeault and topological...
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  • holomorphic vector field of a compact complex manifold to a sum over its Dolbeault cohomology groups. If f is an automorphism of a compact complex manifold M with...
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  • Thumbnail for Jean-Pierre Demailly
    compact complex manifold X {\displaystyle X} , an element of the Dolbeault cohomology group H 1 , 1 ( X , R ) {\displaystyle H^{1,1}(X,\mathbb {R} )} is...
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  • formula relating the ∂ ¯ {\displaystyle {\bar {\partial }}} -Laplacian (see Dolbeault complex) and the Euclidean Laplacian on (p,q)-forms. Specifically, let...
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  • Mathematical Society Intertwining Ladder Representations for SU(p,q) into Dolbeault Cohomology, in Non-Commutative Harmonic Analysis, Progr. Math. 220, Birkhäuser...
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  • Thumbnail for David Mumford
    Kähler metric on a complex manifold, and the Hodge–Lefschetz–Dolbeault theorems on sheaf cohomology break down in every possible way. In the first Pathologies...
    21 KB (2,134 words) - 16:03, 7 July 2025
  • r over X, here H p , q ( X , E ) {\displaystyle H^{p,q}(X,E)} is Dolbeault cohomology group, where Ω X p {\displaystyle \Omega _{X}^{p}} denotes the sheaf...
    10 KB (1,056 words) - 16:39, 23 May 2025
  • Thumbnail for Henri Cartan
    Motivated by the solution to the Cousin problems, he worked on sheaf cohomology and coherent sheaves and proved two powerful results, Cartan's theorems...
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  • 1007/BF01425536. S2CID 122113902. Cartan, H.; Bruhat, F.; Cerf, Jean.; Dolbeault, P.; Frenkel, Jean.; Hervé, Michel; Malatian.; Serre, J-P. "Séminaire...
    11 KB (1,293 words) - 00:04, 8 June 2025
  • Levi form) A differential operator, analogous to the Dolbeault operator, and an associated cohomology (the tangential Cauchy–Riemann complex). Embedded CR...
    36 KB (5,630 words) - 14:42, 16 June 2025
  • {\bar {\partial }}_{A}:L_{1}^{2}(E)\to L^{2}(\Omega ^{0,1}(E))} , the Dolbeault operators of the Chern connections A ∈ A {\displaystyle A\in {\mathcal...
    13 KB (2,063 words) - 13:52, 24 June 2023
  • }}\log \det \left(g_{\alpha {\overline {\beta }}}\right)} where ∂ is the Dolbeault operator and g α β ¯ = g ( ∂ ∂ z α , ∂ ∂ z ¯ β ) . {\displaystyle g_{\alpha...
    34 KB (5,807 words) - 20:11, 6 July 2025
  • called the Dolbeault operators. It follows that ⋆ d f = − i ∂ f + i ∂ ¯ f . {\displaystyle \star df=-i\partial f+i{\bar {\partial }}f.} The Dolbeault operators...
    78 KB (11,073 words) - 22:20, 25 March 2024
  • Les espaces hermitiens symétriques (hermitian symmetric spaces) Pierre Dolbeault, Le théorème de Riemann-Roch sur les surfaces kählériennes compactes,...
    20 KB (2,319 words) - 03:35, 20 March 2024