Hard Lefschetz theorem

E551969

result in Hodge theory theorem

The Hard Lefschetz theorem is a fundamental result in algebraic geometry and Hodge theory that relates the cohomology groups of a compact Kähler manifold via repeated cup product with the Kähler class, yielding powerful symmetry and duality properties.

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All labels observed (3)

Label	Occurrences
Hard Lefschetz theorem canonical	2
Lefschetz theorem on (1,1)-classes	1
hard Lefschetz theorem	1

Statements (46)

Predicate	Object
instanceOf	result in Hodge theory ⓘ theorem ⓘ
appliesTo	compact Kähler manifold ⓘ
assumes	manifold is Kähler ⓘ manifold is compact ⓘ
equivalentTo	certain representation-theoretic sl(2)-actions on cohomology ⓘ
field	Hodge theory NERFINISHED ⓘ algebraic geometry ⓘ complex geometry ⓘ differential geometry ⓘ
generalizedBy	Hard Lefschetz theorem for intersection cohomology NERFINISHED ⓘ
hasVariant	Lefschetz hyperplane theorem NERFINISHED ⓘ weak Lefschetz theorem NERFINISHED ⓘ
historicalContext	developed in the context of Lefschetz’s work on hyperplane sections and Hodge theory ⓘ
holdsFor	smooth projective varieties over the complex numbers ⓘ
implies	Lefschetz decomposition of cohomology NERFINISHED ⓘ constraints on Hodge numbers ⓘ isomorphisms between certain cohomology groups ⓘ symmetry of Betti numbers for compact Kähler manifolds ⓘ
isPartOf	Lefschetz theorems NERFINISHED ⓘ
isRelatedTo	Hodge–Riemann bilinear relations NERFINISHED ⓘ Lefschetz (1,1)-theorem NERFINISHED ⓘ Lefschetz fixed-point theorem NERFINISHED ⓘ Poincaré duality theorem NERFINISHED ⓘ
isUsedIn	Hodge theory of complex manifolds ⓘ intersection cohomology ⓘ mirror symmetry ⓘ representation theory of Lie algebras ⓘ study of perverse sheaves ⓘ study of projective algebraic varieties ⓘ topology of Kähler manifolds ⓘ
relates	cohomology groups in complementary degrees ⓘ cohomology via repeated cup product with the Kähler class ⓘ
requires	existence of a Kähler metric ⓘ finite-dimensional cohomology groups ⓘ
states	for a compact Kähler manifold of complex dimension n, cup product with powers of the Kähler class induces isomorphisms H^{k}(X) → H^{2n-k}(X) NERFINISHED ⓘ
usesConcept	Hodge decomposition NERFINISHED ⓘ Kähler class ⓘ Kähler form NERFINISHED ⓘ Lefschetz operator NERFINISHED ⓘ Poincaré duality NERFINISHED ⓘ cohomology group ⓘ cup product ⓘ de Rham cohomology NERFINISHED ⓘ primitive cohomology ⓘ singular cohomology ⓘ

Referenced by (4)

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

Lefschetz operator → roleIn → Hard Lefschetz theorem ⓘ

Weil cohomology → hasAxiom → Hard Lefschetz theorem ⓘ

this entity surface form: hard Lefschetz theorem

Lefschetz → notableFor → Hard Lefschetz theorem ⓘ

subject surface form: Solomon Lefschetz

this entity surface form: Lefschetz theorem on (1,1)-classes

Lefschetz hyperplane theorem → relatedTo → Hard Lefschetz theorem ⓘ