Standard Conjectures on Algebraic Cycles
E680777
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Standard Conjectures on Algebraic Cycles canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7678370 — 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: Standard Conjectures on Algebraic Cycles Context triple: [Hodge Conjecture, relatedTo, Standard Conjectures on Algebraic Cycles]
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A.
Hodge–Riemann bilinear relations
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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B.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
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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.
Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
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E.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic 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: Standard Conjectures on Algebraic Cycles Target entity description: The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
-
A.
Hodge–Riemann bilinear relations
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.
-
B.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
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.
Topological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry is a foundational mathematical monograph by Friedrich Hirzebruch that applies topological techniques, particularly characteristic classes and cobordism theory, to problems in algebraic geometry.
-
E.
Weil cohomology
Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
conjecture family
ⓘ
conjecture in algebraic geometry ⓘ mathematical conjecture ⓘ |
| aim |
provide foundational theory of algebraic cycles
ⓘ
relate algebraic cycles to cohomology ⓘ underpin theory of pure motives ⓘ |
| appliesTo |
smooth projective varieties
ⓘ
varieties over arbitrary fields ⓘ |
| assumes | existence of Weil cohomology theories ⓘ |
| component |
Hodge type standard conjecture
ⓘ
Künneth type standard conjecture ⓘ Lefschetz type standard conjecture ⓘ numerical equivalence equals homological equivalence conjecture ⓘ |
| concerns |
algebraic cycles modulo homological equivalence
ⓘ
algebraic cycles modulo numerical equivalence ⓘ hard Lefschetz theorem for algebraic cycles NERFINISHED ⓘ positivity properties of intersection forms ⓘ |
| field |
algebraic geometry
ⓘ
arithmetic geometry ⓘ motivic theory ⓘ |
| formulatedInDecade | 1960s ⓘ |
| hasConsequence |
semisimplicity of certain motive categories
ⓘ
standard properties of numerical equivalence ⓘ |
| implies |
Lefschetz decomposition for algebraic cycles
ⓘ
Weil conjectures over finite fields ⓘ existence of certain algebraic correspondences ⓘ symmetry of Betti numbers for smooth projective varieties ⓘ |
| influenced |
development of modern motive theory
ⓘ
research on algebraic K-theory ⓘ work on the Tate conjecture ⓘ |
| involves |
Lefschetz operators
NERFINISHED
ⓘ
algebraic correspondences ⓘ intersection theory ⓘ polarizations on cohomology ⓘ |
| mainTopic |
algebraic cycles
ⓘ
cohomology of algebraic varieties ⓘ theory of motives ⓘ |
| motivation |
construct a semisimple category of pure motives
ⓘ
explain properties of zeta functions of varieties ⓘ |
| namedAfter | algebraic cycles ⓘ |
| openProblemIn |
algebraic geometry
ⓘ
number theory ⓘ |
| proposedBy | Alexander Grothendieck NERFINISHED ⓘ |
| relatedTo |
Grothendieck motives
NERFINISHED
ⓘ
Hodge conjecture NERFINISHED ⓘ Tate conjecture NERFINISHED ⓘ Weil cohomology theory NERFINISHED ⓘ |
| status | open ⓘ |
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Subject: Standard Conjectures on Algebraic Cycles Description of subject: The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.