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Holomorphic vector bundle

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : E → X is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a…

Hermitian metrics on a holomorphic vector bundle, Cohomology of holomorphic vector bundles & Dolbeault operators

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Hermitian metrics on a holomorphic vector bundle

Hermitian metricConnectionE-valued 1-formContractionParallel transport
10 related topics

Cohomology of holomorphic vector bundles

Sheaf cohomologyExact sequencesBaer sumSheaf extensionDolbeault's theorem
6 related topics

Dolbeault operators

Cauchy–Riemann operatorNewlander–Nirenberg theoremRinged spaceTopological manifoldHomotopy operator
5 related topics

The sheaves of forms with values in a holomorphic vector bundle

Tensor productFinePartitions of unityDifferential operatorDolbeault operator
5 related topics

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Overview

Definition through trivialization

The sheaf of holomorphic sections

Basic examples

Dolbeault operators

The sheaves of forms with values in a holomorphic vector bundle

Cohomology of holomorphic vector bundles

The Picard group

Hermitian metrics on a holomorphic vector bundle

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Holomorphic vector bundle

Nodes55
Edges54
Triples26
Avg. degree1.96
Density0.036364
Components1

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Holomorphic vector bundle

Top relations

related to Dolbeault operators · 7
Holomorphic vector bundle → Cauchy, Define, In, Riemann, Suppose, Then, This
related to Cohomology of holomorphic vector bundles · 5
Holomorphic vector bundle → Baer, For, If, In, We
related to Hermitian metrics on a holomorphic vector bundle · 5
Holomorphic vector bundle → Chern, Ex, Indeed, Let, Then
related to The sheaf of holomorphic sections · 4
Holomorphic vector bundle → If, Let, Such, This
related to The sheaves of forms with values in a holomorphic vector bundle · 3
Holomorphic vector bundle → Dolbeault, If, These
is a · 1
Holomorphic vector bundle → complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π
related to External links · 1
Holomorphic vector bundle → Splitting

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Important terminology

holomorphic displaystyle bundle vector complex sheaf manifold bundles mathcal line smooth structure operator partial functions dolbeault defined bar cohomology group

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Holomorphic vector bundleis acomplex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π0.90text
Holomorphic vector bundlerelated to Cohomology of holomorphic vector bundlesIf0.60section
Holomorphic vector bundlerelated to Cohomology of holomorphic vector bundlesIn0.60section
Holomorphic vector bundlerelated to Cohomology of holomorphic vector bundlesWe0.60section
Holomorphic vector bundlerelated to Cohomology of holomorphic vector bundlesFor0.60section
Holomorphic vector bundlerelated to Cohomology of holomorphic vector bundlesBaer0.60section
Holomorphic vector bundlerelated to Dolbeault operatorsSuppose0.60section
Holomorphic vector bundlerelated to Dolbeault operatorsThen0.60section
Holomorphic vector bundlerelated to Dolbeault operatorsIn0.60section
Holomorphic vector bundlerelated to Dolbeault operatorsDefine0.60section
Holomorphic vector bundlerelated to Dolbeault operatorsCauchy0.60section
Holomorphic vector bundlerelated to Dolbeault operatorsRiemann0.60section

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