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 (1)

Label	Occurrences
Lefschetz operator canonical	1

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 ⓘ

Referenced by (1)

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

Kähler manifold → hasOperator → Lefschetz operator ⓘ