Lefschetz operator

E129503

linear operator mathematical concept

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.

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All labels observed (3)

Label	Occurrences
Lefschetz operator canonical	4
Lefschetz operator L	1
Lefschetz operators	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.
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
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 decomposition ⓘ linked to: Hodge theory 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

Referenced by (6)

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

Kähler manifold → hasOperator → Lefschetz operator ⓘ

Hard Lefschetz theorem → usesConcept → Lefschetz operator ⓘ

Kähler identities → relates → Lefschetz operator ⓘ

Kähler identities → involves → Lefschetz operator L ⓘ

linked to: Lefschetz operator

Hodge–Riemann bilinear relations → usesConcept → Lefschetz operator ⓘ

Standard Conjectures on Algebraic Cycles → involves → Lefschetz operators ⓘ