Hodge Conjecture
E173923
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hodge Conjecture canonical | 2 |
| Hodge conjecture | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1523321 — 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 Conjecture Context triple: [Millennium Prize Problem, hasProblem, Hodge Conjecture]
-
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.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
C.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
D.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
E.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra 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: Hodge Conjecture Target entity description: 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.
-
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.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
C.
Riemann hypothesis
The Riemann hypothesis is a famous unsolved conjecture in number theory asserting that all nontrivial zeros of the Riemann zeta function lie on a critical line in the complex plane, with deep implications for the distribution of prime numbers.
-
D.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
-
E.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical conjecture
ⓘ
open problem in mathematics ⓘ problem in algebraic geometry ⓘ |
| asksWhether | every Hodge class is a rational linear combination of classes of algebraic subvarieties ⓘ |
| cohomologyType | singular cohomology with rational coefficients ⓘ |
| concerns |
Hodge classes
ⓘ
Hodge decomposition ⓘ algebraic cycles ⓘ cohomology classes ⓘ rational cohomology ⓘ smooth projective complex varieties ⓘ |
| difficulty | considered extremely difficult ⓘ |
| dimensionWhereNontrivialBegins | complex dimension 3 ⓘ |
| domain | non-singular projective complex varieties ⓘ |
| field |
Hodge theory
ⓘ
algebraic geometry ⓘ complex geometry ⓘ |
| formulatedInTermsOf |
algebraic cycles modulo homological equivalence
ⓘ
rational Hodge structures ⓘ |
| hasPrize |
Millennium Prize Problem
ⓘ
surface form:
Clay Millennium Prize
|
| historicalPeriod | 20th century mathematics ⓘ |
| holdsFor | divisors by Lefschetz (1,1)-theorem ⓘ |
| implies | algebraicity of certain Hodge classes ⓘ |
| importance |
central to the relationship between topology and algebraic geometry
ⓘ
guides research in Hodge theory and algebraic cycles ⓘ |
| involves |
(p,p)-classes in the Hodge decomposition
ⓘ
algebraic cycles of codimension p ⓘ intersection theory ⓘ |
| isSpecialCaseOf | general philosophy relating topology and algebraic geometry ⓘ |
| knownFor | being one of the Clay Millennium Prize Problems ⓘ |
| namedAfter | W. V. D. Hodge ⓘ |
| partiallyProvenFor |
abelian varieties of CM-type in some cases
ⓘ
certain low-dimensional varieties ⓘ |
| predicts |
that certain Hodge classes are algebraic cycles with rational coefficients
ⓘ
which cohomology classes are algebraic ⓘ |
| prizeAmount | 1000000 USD ⓘ |
| recognizedBy | Clay Mathematics Institute ⓘ |
| relatedTo |
Lefschetz hyperplane theorem
ⓘ
surface form:
Lefschetz (1,1)-theorem
Standard Conjectures on Algebraic Cycles ⓘ Tate Conjecture ⓘ Weil conjectures ⓘ
surface form:
Weil Conjectures
|
| status |
unproven in general
ⓘ
unsolved ⓘ |
| varietyCondition |
defined over the complex numbers
ⓘ
projective ⓘ smooth ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Hodge Conjecture Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Hodge conjecture