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 (7)

Label	Occurrences
Hard Lefschetz theorem canonical	4
hard Lefschetz theorem	2
Hard Lefschetz theorem for intersection cohomology	1
Lefschetz theorem on (1,1)-classes	1
Lefschetz theorems	1
hard Lefschetz theorem for algebraic cycles	1
strong Lefschetz hyperplane theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T5837328 — 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: Hard Lefschetz theorem
Context triple: [Lefschetz operator, roleIn, Hard Lefschetz theorem]

A. 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.
B. Lefschetz
Lefschetz is a surname most notably associated with Solomon Lefschetz, a pioneering mathematician in algebraic topology and geometry.
C. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
D. Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
E. Lefschetz pencil
A Lefschetz pencil is a geometric structure on an algebraic variety given by a one-parameter family of hyperplane sections with only isolated, well-controlled singularities, fundamental in the study of its topology and geometry.
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: Hard Lefschetz theorem
Target entity description: 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.

A. 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.
B. Lefschetz
Lefschetz is a surname most notably associated with Solomon Lefschetz, a pioneering mathematician in algebraic topology and geometry.
C. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
D. Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
E. Lefschetz pencil
A Lefschetz pencil is a geometric structure on an algebraic variety given by a one-parameter family of hyperplane sections with only isolated, well-controlled singularities, fundamental in the study of its topology and geometry.
F. None of above. chosen

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 ⓘ algebraic geometry ⓘ complex geometry ⓘ differential geometry ⓘ
generalizedBy	Hard Lefschetz theorem for intersection cohomology ⓘ linked to: Hard Lefschetz theorem
hasVariant	Lefschetz hyperplane theorem ⓘ weak Lefschetz theorem ⓘ linked to: Lefschetz hyperplane theorem
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 ⓘ linked to: Lefschetz decomposition constraints on Hodge numbers ⓘ isomorphisms between certain cohomology groups ⓘ symmetry of Betti numbers for compact Kähler manifolds ⓘ
isPartOf	Lefschetz theorems ⓘ linked to: Hard Lefschetz theorem
isRelatedTo	Hodge–Riemann bilinear relations ⓘ Lefschetz (1,1)-theorem ⓘ Lefschetz fixed-point theorem ⓘ Poincaré duality theorem ⓘ linked to: Poincaré duality
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) ⓘ
usesConcept	Hodge decomposition ⓘ Kähler class ⓘ Kähler form ⓘ Lefschetz operator ⓘ Poincaré duality ⓘ cohomology group ⓘ cup product ⓘ de Rham cohomology ⓘ primitive cohomology ⓘ singular cohomology ⓘ

How these facts were elicited