Hodge–Riemann bilinear relations
E551973
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hodge–Riemann bilinear relations canonical | 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.
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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.
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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
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
Hodge–Riemann relations in combinatorial Hodge theory NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
hard Lefschetz theorem NERFINISHED ⓘ signature properties of intersection pairings ⓘ |
| motivated |
Hodge–Riemann relations for polytopes and matroids
NERFINISHED
ⓘ
generalizations to mixed Hodge structures ⓘ |
| namedAfter |
Bernhard Riemann
NERFINISHED
ⓘ
W. V. D. Hodge NERFINISHED ⓘ |
| property |
definiteness of the intersection form on primitive classes
ⓘ
orthogonality of different primitive components ⓘ positivity on primitive cohomology ⓘ |
| relatedTo |
Kähler identities
ⓘ
Lefschetz decomposition NERFINISHED ⓘ Weil conjectures NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Lefschetz operator NERFINISHED ⓘ intersection form on cohomology ⓘ primitive cohomology ⓘ |
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Input
Subject: Hodge–Riemann bilinear relations Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.