Hard Lefschetz theorem
E551969
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Hard Lefschetz theorem canonical | 2 |
| Lefschetz theorem on (1,1)-classes | 1 |
| hard Lefschetz 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.
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B.
Lefschetz
Lefschetz is a surname most notably associated with Solomon Lefschetz, a pioneering mathematician in algebraic topology and geometry.
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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.
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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
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hard Lefschetz theorem Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
hard Lefschetz theorem
subject surface form:
Solomon Lefschetz
this entity surface form:
Lefschetz theorem on (1,1)-classes