Kähler identities

E551970

mathematical concept result in Kähler geometry result in complex geometry result in differential geometry

Kähler identities are fundamental commutation relations in Kähler geometry that link the Lefschetz operator, its adjoint, and the Dolbeault operators, playing a key role in Hodge theory and complex differential geometry.

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All labels observed (2)

Label	Occurrences
Kähler geometry	1
Kähler identities canonical	1

Statements (50)

Predicate	Object
instanceOf	mathematical concept ⓘ result in Kähler geometry ⓘ result in complex geometry ⓘ result in differential geometry ⓘ
appearsIn	textbooks on Hodge theory ⓘ textbooks on Kähler manifolds ⓘ textbooks on complex geometry ⓘ
appliesTo	Kähler manifold ⓘ compact Kähler manifold ⓘ differential forms on a Kähler manifold ⓘ
context	complex manifolds with Kähler metric ⓘ
expresses	commutation relations between L and \\bar{∂} ⓘ commutation relations between L and Λ ⓘ commutation relations between L and ∂ ⓘ commutation relations between Λ and \\bar{∂} ⓘ commutation relations between Λ and ∂ ⓘ
field	Hodge theory ⓘ Kähler geometry ⓘ algebraic geometry ⓘ complex differential geometry ⓘ global analysis ⓘ
historicalPeriod	20th century mathematics ⓘ
implies	Δ_d = 2Δ_∂ = 2Δ_\\bar{∂} on Kähler manifolds ⓘ
involves	Dolbeault Laplacian NERFINISHED ⓘ Hermitian metric ⓘ Hodge Laplacian NERFINISHED ⓘ Hodge star operator ⓘ Kähler form NERFINISHED ⓘ Laplace operator ⓘ Lefschetz operator L NERFINISHED ⓘ Riemannian metric ⓘ adjoint operator Λ ⓘ complex structure ⓘ
keyRoleIn	Hard Lefschetz theorem NERFINISHED ⓘ Hodge decomposition NERFINISHED ⓘ Hodge theory on Kähler manifolds ⓘ Lefschetz decomposition ⓘ proof of Hodge symmetry ⓘ proof of ∂∂̄-lemma ⓘ
namedAfter	Erich Kähler NERFINISHED ⓘ
relates	Dolbeault operator \\bar{∂} NERFINISHED ⓘ Dolbeault operator ∂ NERFINISHED ⓘ Lefschetz operator NERFINISHED ⓘ adjoint Dolbeault operator \\bar{∂}* ⓘ adjoint Dolbeault operator ∂* ⓘ adjoint Lefschetz operator ⓘ
usedFor	computing cohomology of Kähler manifolds ⓘ identification of Laplacians Δ_d, Δ_∂, Δ_\\bar{∂} ⓘ relating de Rham and Dolbeault cohomology ⓘ showing harmonic forms decompose into (p,q)-types ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Lefschetz operator → relatedConcept → Kähler identities ⓘ

Hodge theory → fieldOfStudy → Kähler identities ⓘ

this entity surface form: Kähler geometry