Kähler manifold

E23190

Riemannian manifold complex manifold geometric structure symplectic manifold

A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.

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All labels observed (4)

Label	Occurrences
Kähler manifold canonical	3
Kähler manifolds	3
Kähler geometry	2
Kähler metric	1

How this entity was disambiguated

This entity first appeared as the object of triple T179366 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Kähler manifold
Context triple: [Riemannian manifold, hasVariant, Kähler manifold]

A. Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
B. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
C. Ricci curvature tensor
The Ricci curvature tensor is a geometric object in differential geometry that measures how volumes in a curved space-time deviate from those in flat space, playing a central role in general relativity.
D. Riemann curvature tensor
The Riemann curvature tensor is a fundamental geometric object in differential geometry that measures how much a Riemannian manifold deviates from being flat by encoding the intrinsic curvature of the space.
E. Kretschmann scalar
The Kretschmann scalar is a curvature invariant in general relativity that combines components of the Riemann tensor into a single scalar quantity used to characterize the intensity of spacetime curvature, especially near singularities.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Kähler manifold
Target entity description: A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.

A. Riemannian manifolds
Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
B. Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
C. Ricci curvature tensor
The Ricci curvature tensor is a geometric object in differential geometry that measures how volumes in a curved space-time deviate from those in flat space, playing a central role in general relativity.
D. Riemann curvature tensor
The Riemann curvature tensor is a fundamental geometric object in differential geometry that measures how much a Riemannian manifold deviates from being flat by encoding the intrinsic curvature of the space.
E. Kretschmann scalar
The Kretschmann scalar is a curvature invariant in general relativity that combines components of the Riemann tensor into a single scalar quantity used to characterize the intensity of spacetime curvature, especially near singularities.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	Riemannian manifold ⓘ complex manifold ⓘ geometric structure ⓘ symplectic manifold ⓘ
associatedForm	Kähler form ⓘ fundamental 2-form ⓘ
definitionCondition	J is orthogonal with respect to the Riemannian metric ⓘ Kähler form is d-closed ⓘ ∇J = 0 for Levi-Civita connection ∇ ⓘ
dimension	complex dimension n ⓘ even real dimension ⓘ
field	algebraic geometry ⓘ complex geometry ⓘ differential geometry ⓘ symplectic geometry ⓘ
hasCohomologyProperty	Betti numbers satisfy b_{2k+1} is even ⓘ admits Hodge decomposition ⓘ satisfies Hodge symmetry ⓘ satisfies hard Lefschetz theorem ⓘ
hasCurvatureProperty	Riemann curvature tensor has Kähler symmetries ⓘ
hasExample	Calabi–Yau manifold ⓘ Riemann surface with any Hermitian metric ⓘ complex projective space CP^n with Fubini–Study metric ⓘ complex tori with flat metric ⓘ smooth projective algebraic variety over C ⓘ
hasOperator	Dolbeault operators ∂ and ∂̄ ⓘ Lefschetz operator ⓘ
hasProperty	Kähler form is closed ⓘ Kähler identities hold between ∂, ∂̄, and Lefschetz operators ⓘ Laplace–Beltrami operator equals Hodge Laplacian on forms ⓘ Levi-Civita connection preserves complex structure ⓘ complex structure is integrable ⓘ holonomy group is contained in U(n) ⓘ metric is Hermitian with respect to complex structure ⓘ symplectic form is compatible with complex structure ⓘ
hasStructure	Hermitian metric ⓘ Riemannian metric ⓘ complex structure ⓘ symplectic form ⓘ
implies	underlying manifold is Riemannian ⓘ underlying manifold is complex ⓘ underlying manifold is symplectic ⓘ
localCoordinateDescription	Kähler form is i∂∂̄ of a real-valued potential function (locally) ⓘ metric is given by a Kähler potential ⓘ
namedAfter	Erich Kähler ⓘ
usedIn	Hodge theory ⓘ algebraic geometry ⓘ string theory ⓘ

How these facts were elicited

Referenced by (9)

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

Riemannian manifold → hasVariant → Kähler manifold ⓘ

subject linked to: Riemannian manifolds

Ricci flow → appliedIn → Kähler geometry ⓘ

linked to: Kähler manifold

Kähler form → usedIn → Kähler geometry ⓘ

linked to: Kähler manifold

Kähler–Ricci flow → definedOn → Kähler manifold ⓘ

Erich Kähler → knownFor → Kähler manifolds ⓘ

linked to: Kähler manifold

Erich Kähler → notableConcept → Kähler manifold ⓘ

Erich Kähler → notableConcept → Kähler metric ⓘ

linked to: Kähler manifold

Bochner–Kodaira–Nakano identity → appliesTo → Kähler manifolds ⓘ

linked to: Kähler manifold

Kähler geometry → studies → Kähler manifolds ⓘ

linked to: Kähler manifold

All labels observed (4)

How this entity was disambiguated Show

Statements (48)

How these facts were elicited Show

Referenced by (9)

How this entity was disambiguated

How these facts were elicited