Lefschetz operator
E129503
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lefschetz operator canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1138357 — 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: Lefschetz operator Context triple: [Kähler manifold, hasOperator, Lefschetz operator]
-
A.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
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B.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
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C.
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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D.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
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E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- 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: Lefschetz operator Target entity description: The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
-
A.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
B.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
C.
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.
-
D.
Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
-
E.
Riemann–Hurwitz formula
The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex analysis that relates the genera of two Riemann surfaces connected by a branched covering map, accounting for the ramification data.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
linear operator
ⓘ
mathematical concept ⓘ |
| actsOn | differential forms ⓘ |
| analog | cup product with the hyperplane class in projective space ⓘ |
| appearsIn | Hard Lefschetz isomorphisms L^{k}: H^{n-k}(X) → H^{n+k}(X) ⓘ |
| appliesTo |
Dolbeault cohomology classes
ⓘ
cohomology groups of compact Kähler manifolds ⓘ |
| codomain | space of differential forms on a Kähler manifold ⓘ |
| construction | given a Kähler manifold (X, ω), L(α) = ω ∧ α ⓘ |
| context |
cohomology ring of a compact Kähler manifold
ⓘ
intersection theory on projective varieties ⓘ |
| definition | wedge product with the Kähler form ⓘ |
| dependsOn |
Kähler form
ⓘ
Riemannian metric compatible with the complex structure ⓘ complex structure of the manifold ⓘ |
| domain | space of differential forms on a Kähler manifold ⓘ |
| field |
Hodge theory
ⓘ
Kähler geometry ⓘ algebraic geometry ⓘ differential geometry ⓘ |
| generalizationOf | cup product with an ample class in algebraic geometry ⓘ |
| mathematicalDomain |
complex geometry
ⓘ
representation theory of Lie algebras ⓘ topology ⓘ |
| namedAfter | Solomon Lefschetz ⓘ |
| property |
commutes with the Laplacian on a Kähler manifold
ⓘ
is a bounded operator on spaces of harmonic forms of fixed degree ⓘ is ℂ-linear on complex differential forms ⓘ is ℝ-linear on real differential forms ⓘ raises the degree of a differential form by 2 ⓘ |
| relatedConcept |
Hodge theory
ⓘ
surface form:
Hodge decomposition
Kähler identities ⓘ adjoint Lefschetz operator ⓘ primitive cohomology ⓘ |
| roleIn |
Hard Lefschetz theorem
ⓘ
Hodge theory on Kähler manifolds ⓘ Lefschetz decomposition ⓘ representation of sl(2,ℂ) on cohomology ⓘ |
| satisfies | sl(2)-commutation relations with its adjoint and the grading operator ⓘ |
| symbol | L ⓘ |
| usedIn |
construction of primitive decomposition of cohomology
ⓘ
proof of the Hard Lefschetz theorem ⓘ proof of the Hodge–Riemann bilinear relations ⓘ |
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: Lefschetz operator Description of subject: The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.