Deligne cohomology
E269189
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Deligne cohomology canonical | 1 |
| Deligne complex | 1 |
| relative Deligne cohomology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2437616 — 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: Deligne cohomology Context triple: [Pierre Deligne, knownFor, Deligne cohomology]
-
A.
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.
-
B.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
C.
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.
-
D.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
-
E.
étale cohomology
Étale cohomology is a cohomology theory in algebraic geometry that allows one to apply topological and cohomological methods to schemes, particularly over fields with nontrivial arithmetic such as finite fields.
- 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: Deligne cohomology Target entity description: 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.
-
A.
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.
-
B.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
C.
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.
-
D.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
-
E.
étale cohomology
Étale cohomology is a cohomology theory in algebraic geometry that allows one to apply topological and cohomological methods to schemes, particularly over fields with nontrivial arithmetic such as finite fields.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology theory
ⓘ
mathematical concept ⓘ refined cohomology theory ⓘ |
| appliesTo |
complex algebraic varieties
ⓘ
complex manifolds ⓘ smooth complex algebraic varieties ⓘ |
| captures |
arithmetic information
ⓘ
topological information ⓘ |
| combines |
differential forms
ⓘ
singular cohomology ⓘ |
| compatibleWith |
Hodge filtration
ⓘ
mixed Hodge structures on cohomology ⓘ |
| constructedUsing |
Deligne cohomology
self-linksurface differs
ⓘ
surface form:
Deligne complex
sheaf cohomology ⓘ truncated de Rham complex ⓘ |
| context | complex algebraic varieties over the complex numbers ⓘ |
| encodes |
information about algebraic cycles
ⓘ
information about differential characters ⓘ |
| field |
Hodge theory
ⓘ
algebraic geometry ⓘ arithmetic geometry ⓘ complex geometry ⓘ |
| generalizes | Picard group with connection ⓘ |
| hasCoefficientType |
integral coefficients
ⓘ
rational coefficients ⓘ real coefficients ⓘ |
| hasVariant |
absolute Deligne cohomology
ⓘ
integral Deligne cohomology ⓘ rational Deligne cohomology ⓘ real Deligne cohomology ⓘ Deligne cohomology self-linksurface differs ⓘ
surface form:
relative Deligne cohomology
|
| introducedBy | Pierre Deligne ⓘ |
| namedAfter | Pierre Deligne ⓘ |
| refines |
de Rham cohomology
ⓘ
singular cohomology with coefficients ⓘ |
| relatedTo |
Beilinson regulator
ⓘ
surface form:
Beilinson regulators
Cheeger–Simons differential characters ⓘ Chow groups ⓘ Hodge realization of motives ⓘ K-theory ⓘ
surface form:
algebraic K-theory
cycle class maps ⓘ differential cohomology theories ⓘ intermediate Jacobians ⓘ mixed Hodge structures ⓘ regulator maps ⓘ |
| usedFor |
defining regulator maps from K-theory
ⓘ
describing higher gerbes with connection ⓘ describing line bundles with connection ⓘ refining Chern classes ⓘ |
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: Deligne cohomology Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
relative Deligne cohomology
this entity surface form:
Deligne complex