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.
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
NERFINISHED
ⓘ
mixed Hodge structure ⓘ |
| field |
Hodge theory
NERFINISHED
ⓘ
algebraic geometry ⓘ complex geometry ⓘ |
| generalizedBy | filtrations in p-adic Hodge theory ⓘ |
| introducedInContextOf | Hodge theory of complex algebraic varieties NERFINISHED ⓘ |
| isDecreasing | true ⓘ |
| mathematicalDomain |
cohomological algebra
ⓘ
complex algebraic geometry ⓘ |
| namedAfter | W. V. D. Hodge NERFINISHED ⓘ |
| property |
exhaustive filtration
ⓘ
finite filtration ⓘ separated filtration ⓘ |
| relatedConcept |
Griffiths transversality
NERFINISHED
ⓘ
Hodge numbers ⓘ Hodge structure NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.