Hodge–Riemann bilinear relations

E551973

mathematical theorem result in Hodge theory

The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.

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All labels observed (5)

Label	Occurrences
Hodge–Riemann bilinear relations canonical	2
Hodge index theorem	1
Hodge–Riemann relations for intersection cohomology	1
Hodge–Riemann relations for polytopes and matroids	1
Hodge–Riemann relations in combinatorial Hodge theory	1

How this entity was disambiguated

This entity first appeared as the object of triple T5837375 — 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: Hodge–Riemann bilinear relations
Context triple: [Hodge theory, studies, Hodge–Riemann bilinear relations]

A. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
B. Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
C. Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
D. Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
E. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
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: Hodge–Riemann bilinear relations
Target entity description: The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.

A. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
B. Grothendieck–Ogg–Shafarevich formula
The Grothendieck–Ogg–Shafarevich formula is a result in arithmetic geometry that relates the Euler characteristic of an ℓ-adic sheaf on a curve over a finite field to local invariants such as conductors and ramification data.
C. Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
D. Brill–Noether theory
Brill–Noether theory is a branch of algebraic geometry that studies linear series on algebraic curves, particularly the existence and dimension of spaces of special divisors and maps to projective spaces.
E. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in Hodge theory ⓘ
appearsIn	classical Hodge theory of compact Kähler manifolds ⓘ modern treatments of algebraic geometry textbooks ⓘ
appliesTo	compact Kähler manifolds ⓘ smooth projective varieties over the complex numbers ⓘ
assumes	existence of a Kähler metric ⓘ finite-dimensional cohomology groups ⓘ
concerns	Hermitian form induced by Kähler class ⓘ bilinear form on cohomology groups ⓘ
describes	orthogonality properties of intersection forms ⓘ positivity properties of intersection forms ⓘ
field	Hodge theory ⓘ algebraic geometry ⓘ complex geometry ⓘ
formalizes	positivity of the cup product with powers of a Kähler class ⓘ signature behavior of the intersection form on primitive subspaces ⓘ
generalizedBy	Hodge–Riemann relations for intersection cohomology ⓘ linked to: Hodge–Riemann bilinear relations Hodge–Riemann relations in combinatorial Hodge theory ⓘ linked to: Hodge–Riemann bilinear relations
gives	orthogonal decomposition of cohomology into primitive parts ⓘ sign constraints on intersection pairings ⓘ
historicalContext	developed in the 20th century ⓘ
holdsIn	cohomology with complex coefficients ⓘ middle-degree cohomology ⓘ
implies	Hodge index theorem ⓘ linked to: Hodge–Riemann bilinear relations hard Lefschetz theorem ⓘ linked to: Hard Lefschetz theorem signature properties of intersection pairings ⓘ
motivated	Hodge–Riemann relations for polytopes and matroids ⓘ linked to: Hodge–Riemann bilinear relations generalizations to mixed Hodge structures ⓘ
namedAfter	Bernhard Riemann ⓘ W. V. D. Hodge ⓘ
property	definiteness of the intersection form on primitive classes ⓘ orthogonality of different primitive components ⓘ positivity on primitive cohomology ⓘ
relatedTo	Kähler identities ⓘ Lefschetz decomposition ⓘ Weil conjectures ⓘ
role	foundational tool in Kähler geometry ⓘ key ingredient in proofs of Lefschetz-type theorems ⓘ
usedIn	proofs of inequalities for intersection numbers ⓘ study of ample line bundles ⓘ study of the Kähler cone ⓘ study of the topology of algebraic varieties ⓘ
usesConcept	Hodge decomposition ⓘ Lefschetz operator ⓘ intersection form on cohomology ⓘ primitive cohomology ⓘ

How these facts were elicited

Referenced by (6)

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

Hodge theory → studies → Hodge–Riemann bilinear relations ⓘ

Hard Lefschetz theorem → isRelatedTo → Hodge–Riemann bilinear relations ⓘ

Hodge–Riemann bilinear relations → implies → Hodge index theorem ⓘ

linked to: Hodge–Riemann bilinear relations

Hodge–Riemann bilinear relations → motivated → Hodge–Riemann relations for polytopes and matroids ⓘ

linked to: Hodge–Riemann bilinear relations

Hodge–Riemann bilinear relations → generalizedBy → Hodge–Riemann relations in combinatorial Hodge theory ⓘ

linked to: Hodge–Riemann bilinear relations

Hodge–Riemann bilinear relations → generalizedBy → Hodge–Riemann relations for intersection cohomology ⓘ

linked to: Hodge–Riemann bilinear relations

All labels observed (5)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (6)

How this entity was disambiguated

How these facts were elicited