Hodge filtration
E551972
The Hodge filtration is a decreasing sequence of complex subspaces on the cohomology of a complex algebraic variety that encodes its Hodge decomposition and mixed Hodge structure.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hodge filtration canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5837371 — 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 filtration Context triple: [Hodge theory, studies, Hodge filtration]
-
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.
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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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.
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D.
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.
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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 filtration Target entity description: The Hodge filtration is a decreasing sequence of complex subspaces on the cohomology of a complex algebraic variety that encodes its Hodge decomposition and mixed Hodge structure.
-
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.
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.
-
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.
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.
-
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 |
decreasing filtration
ⓘ
filtration ⓘ mathematical concept ⓘ structure in Hodge theory ⓘ |
| appearsIn |
Deligne’s theory of mixed Hodge structures
ⓘ
study of period maps ⓘ theory of algebraic cycles ⓘ variation of Hodge structure ⓘ |
| appliesTo |
cohomology of complex algebraic varieties
ⓘ
de Rham cohomology ⓘ singular cohomology with complex coefficients ⓘ |
| associatedTo |
mixed Hodge structure on cohomology
ⓘ
smooth projective complex variety ⓘ |
| coincidesWith | Hodge–de Rham filtration in the smooth projective case ⓘ |
| compatibleWith | Hodge decomposition H^n(X,ℂ)=⊕_{p+q=n} H^{p,q}(X) ⓘ |
| componentOf |
mixed Hodge structure
ⓘ
pure Hodge structure ⓘ |
| definedBy | F^p H^n(X,ℂ)=⊕_{r≥p} H^{r,n-r}(X) for pure Hodge structures ⓘ |
| definedOn |
cohomology group H^n(X,ℂ)
ⓘ
complex vector space ⓘ |
| determines | Hodge decomposition together with its complex conjugate ⓘ |
| encodes |
Hodge decomposition
ⓘ
mixed Hodge structure ⓘ |
| field |
Hodge theory
ⓘ
algebraic geometry ⓘ complex geometry ⓘ |
| generalizedBy | filtrations in p-adic Hodge theory ⓘ |
| introducedInContextOf |
Hodge theory of complex algebraic varieties
ⓘ
linked to:
Hodge theory
|
| isDecreasing | true ⓘ |
| mathematicalDomain |
cohomological algebra
ⓘ
complex algebraic geometry ⓘ |
| namedAfter | W. V. D. Hodge ⓘ |
| property |
exhaustive filtration
ⓘ
finite filtration ⓘ separated filtration ⓘ |
| relatedConcept |
Griffiths transversality
ⓘ
Hodge numbers ⓘ Hodge structure ⓘ mixed Hodge structure ⓘ |
| relatedStructure | weight filtration ⓘ |
| symbol | F^p H^n(X,ℂ) ⓘ |
| usedFor |
defining mixed Hodge structures
ⓘ
describing Hodge numbers ⓘ formulating Hodge decomposition ⓘ studying variations of Hodge structure ⓘ |
| usedIn |
definition of period domains
ⓘ
study of degenerations of Hodge structures ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hodge filtration Description of subject: The Hodge filtration is a decreasing sequence of complex subspaces on the cohomology of a complex algebraic variety that encodes its Hodge decomposition and mixed Hodge structure.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.