Kähler cone
E551967
The Kähler cone is the convex cone in the cohomology of a complex manifold consisting of classes that can be represented by Kähler forms, encoding its possible Kähler metrics and playing a central role in complex and algebraic geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kähler cone canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5837283 — 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 cone Context triple: [Calabi–Yau manifold, hasInvariant, Kähler cone]
-
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.
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B.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
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C.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
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D.
Calabi–Yau manifold
A Calabi–Yau manifold is a special type of complex manifold with vanishing first Chern class that plays a central role in string theory compactifications and complex algebraic geometry.
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E.
Lefschetz hyperplane theorem
The Lefschetz hyperplane theorem is a fundamental result in algebraic geometry and topology that relates the topology (especially homology and homotopy groups) of a smooth projective variety to that of its hyperplane sections.
- 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 cone Target entity description: The Kähler cone is the convex cone in the cohomology of a complex manifold consisting of classes that can be represented by Kähler forms, encoding its possible Kähler metrics and playing a central role in complex and algebraic geometry.
-
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.
Kähler form
A Kähler form is a closed, positive-definite (1,1)-form that defines the compatible symplectic and Hermitian structure on a Kähler manifold.
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C.
Kähler–Ricci flow
Kähler–Ricci flow is a geometric evolution equation that deforms Kähler metrics on complex manifolds according to their Ricci curvature, playing a central role in complex differential geometry and the study of canonical metrics.
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D.
Calabi–Yau manifold
A Calabi–Yau manifold is a special type of complex manifold with vanishing first Chern class that plays a central role in string theory compactifications and complex algebraic geometry.
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E.
Lefschetz hyperplane theorem
The Lefschetz hyperplane theorem is a fundamental result in algebraic geometry and topology that relates the topology (especially homology and homotopy groups) of a smooth projective variety to that of its hyperplane sections.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
convex cone
ⓘ
geometric object ⓘ subset of cohomology ⓘ |
| appearsIn |
Hodge theory
ⓘ
Yau's solution of the Calabi conjecture (as space of Kähler classes) ⓘ global Torelli theorems for K3 surfaces NERFINISHED ⓘ |
| characterizedBy |
classes representable by closed real (1,1)-forms
ⓘ
classes representable by positive definite (1,1)-forms ⓘ |
| consistsOf | cohomology classes of Kähler forms ⓘ |
| definedOn | complex manifold ⓘ |
| dependsOn |
Hodge decomposition of the manifold
ⓘ
complex structure of the manifold ⓘ |
| encodes | possible Kähler metrics on a complex manifold ⓘ |
| field |
algebraic geometry
ⓘ
complex geometry ⓘ |
| hasBoundaryDescribedBy |
classes of nef but not Kähler divisors (projective case)
ⓘ
degeneration of Kähler metrics ⓘ |
| invariantUnder | biholomorphisms ⓘ |
| is | set of cohomology classes [ω] with ω a Kähler form ⓘ |
| livesIn |
H^{1,1}(X,ℝ)
ⓘ
real (1,1)-cohomology ⓘ |
| mayBeTrivialIf | manifold is non-Kähler ⓘ |
| nonEmptyIf | manifold is Kähler ⓘ |
| property |
convex
ⓘ
open cone ⓘ salient cone ⓘ |
| relatedConcept |
Mori cone
ⓘ
ample cone ⓘ nef cone ⓘ pseudo-effective cone ⓘ |
| relatedTo |
Kähler class
ⓘ
Kähler form ⓘ Kähler metric ⓘ |
| structure | open convex cone in a finite-dimensional real vector space ⓘ |
| studiedBy |
Claire Voisin
NERFINISHED
ⓘ
Jean-Pierre Demailly NERFINISHED ⓘ Shigefumi Mori NERFINISHED ⓘ Vladimir Shokurov NERFINISHED ⓘ |
| subsetOf |
nef cone (for projective manifolds, interior)
ⓘ
positive cone ⓘ |
| usedIn |
Calabi–Yau geometry
NERFINISHED
ⓘ
birational geometry of projective varieties ⓘ classification of compact Kähler manifolds ⓘ minimal model program (via nef and movable cones) ⓘ mirror symmetry ⓘ study of moduli of Kähler metrics ⓘ |
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 cone Description of subject: The Kähler cone is the convex cone in the cohomology of a complex manifold consisting of classes that can be represented by Kähler forms, encoding its possible Kähler metrics and playing a central role in complex and algebraic geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.