Kähler form
E129501
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kähler form canonical | 6 |
| Kähler current | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1138339 — 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 form Context triple: [Kähler manifold, associatedForm, Kähler form]
-
A.
Kähler 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.
-
B.
Cartan structure equations
Cartan structure equations are fundamental differential geometric relations that express curvature and torsion in terms of connection 1-forms on a manifold.
-
C.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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D.
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.
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E.
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.
- 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 form Target entity description: A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
-
A.
Kähler 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.
-
B.
Cartan structure equations
Cartan structure equations are fundamental differential geometric relations that express curvature and torsion in terms of connection 1-forms on a manifold.
-
C.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
(1,1)-form
ⓘ
Hermitian form ⓘ differential form ⓘ geometric structure ⓘ symplectic form ⓘ |
| appearsOn | complex projective space ⓘ |
| associatedWith |
Kähler metric
ⓘ
Kähler structure ⓘ |
| cohomologyClassLiesIn | H^2(M, ℝ) ⓘ |
| compatibleWith |
Hermitian metric
ⓘ
Riemannian metric ⓘ complex structure ⓘ |
| definedOn | Kähler manifold ⓘ |
| defines |
Hermitian structure
ⓘ
Riemannian metric ⓘ symplectic structure ⓘ |
| definesCohomologyClass | Kähler class ⓘ |
| determines |
Levi-Civita connection
ⓘ
volume form ⓘ |
| existsIf | manifold is Kähler ⓘ |
| generalizedBy |
Kähler current
ⓘ
linked to:
Kähler form
|
| hasProperty |
J-invariant
ⓘ
closed ⓘ non-degenerate ⓘ positive-definite ⓘ |
| hasType | real (1,1)-form ⓘ |
| implies |
metric is Hermitian
ⓘ
underlying manifold is complex ⓘ underlying manifold is symplectic ⓘ |
| isClosedUnder | exterior derivative ⓘ |
| isExampleOf | closed positive (1,1)-current (smooth case) ⓘ |
| isInvariantUnder |
complex structure J
ⓘ
holomorphic isometries ⓘ |
| isLocalExpression | i g_{jar{k}} dz^j ∧ dar{z}^k ⓘ |
| isParallelWithRespectTo | Levi-Civita connection ⓘ |
| isRealPartOf | Hermitian form ⓘ |
| isRepresentativeOf |
(1,1)-Dolbeault cohomology class
ⓘ
de Rham cohomology class ⓘ |
| livesIn | Dolbeault type (1,1) ⓘ |
| nonDegenerateOn | tangent bundle ⓘ |
| relatedTo | Fubini–Study form ⓘ |
| satisfies |
dω = 0
ⓘ
ω(X,Y) = g(JX,Y) ⓘ |
| usedIn |
Hodge theory
ⓘ
Kähler geometry ⓘ
linked to:
Kähler manifold
algebraic geometry ⓘ complex differential geometry ⓘ mirror symmetry ⓘ string theory compactifications ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Kähler form Description of subject: A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Kähler form