Differential forms | Complex manifolds

Complex differential form

In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry, Kähler geometry, and Hodge theory. Over non-complex manifolds, they also play a role in the study of almost complex structures, the theory of spinors, and CR structures. Typically, complex forms are considered because of some desirable decomposition that the forms admit. On a complex manifold, for instance, any complex k-form can be decomposed uniquely into a sum of so-called (p,q)-forms: roughly, wedges of p differentials of the holomorphic coordinates with q differentials of their complex conjugates. The ensemble of (p,q)-forms becomes the primitive object of study, and determines a finer geometrical structure on the manifold than the k-forms. Even finer structures exist, for example, in cases where Hodge theory applies. (Wikipedia).

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Complex Numbers for ODEs (1 of 4)

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From playlist Differential Equations

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Differential Equations: Complex Roots of the Characteristic Equation

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From playlist Differential Equations

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From playlist Differential Equations

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From playlist Solve Differential Equation (Particular Solution) #Integration

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Introduction to Differential Equations

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From playlist Differential Equations

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Second order differential equation: complex roots

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From playlist Differential equations

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How to determine the general solution to a differential equation

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From playlist Differential Equations

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From playlist Popular Questions

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How to solve a separable differential equation

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From playlist Solve Differential Equation (Particular Solution) #Integration

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From playlist Mathematics

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From playlist Global Noncommutative Geometry Seminar (Europe)

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Pre-recorded lecture 8: Differentially non-degenerate singular points and global theorems

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From playlist MATRIX-SMRI Symposium: Nijenhuis Geometry companion lectures (Sino-Russian Mathematical Centre)

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John Terilla : A collection of Homotopy Random Variables

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From playlist Geometry

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Francis BROWN - Graph Complexes, Invariant Differential Forms and Feynman integrals

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Georg Tamme: Differential algebraic K theory

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From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"

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Pierre Albin: The sub-Riemannian limit of a contact manifold

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From playlist Global Noncommutative Geometry Seminar (Americas)

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From playlist Mathematics

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Kevin Buzzard (lecture 19/20) Automorphic Forms And The Langlands Program [2017]

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From playlist MSRI Summer School: Automorphic Forms And The Langlands Program, by Kevin Buzzard [2017]

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Differential Equations: Exact DEs Example 1

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From playlist Differential Equations

Related pages

Star domain | Differential form | Exterior derivative | Coordinate system | Ddbar lemma | Complex manifold | Hodge theory | Spinor | De Rham cohomology | Mathematics | Cauchy–Riemann equations | Dolbeault cohomology | Algebraic geometry | Sheaf (mathematics) | Tensor | Vector bundle | Holomorphic function | Manifold | Kähler manifold | Complex number | Differential geometry | Differential of the first kind | Frölicher spectral sequence