mixed Hodge structures
E921615
Mixed Hodge structures are algebraic structures on cohomology groups that generalize pure Hodge structures by incorporating a weight filtration, allowing the study of varieties with singularities or non-compactness.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| mixed Hodge structure | 0 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
structure in Hodge theory ⓘ |
| allowsStudyOf |
non-compact varieties
ⓘ
varieties with singularities ⓘ |
| appearsIn |
Deligne’s mixed Hodge theory
NERFINISHED
ⓘ
study of cohomology with compact support ⓘ study of intersection cohomology ⓘ |
| appliesTo |
cohomology of complex algebraic varieties
ⓘ
cohomology of open varieties ⓘ cohomology of singular varieties ⓘ |
| categoryProperty |
closed under extensions
ⓘ
forms an abelian category ⓘ |
| definedOn | cohomology group ⓘ |
| definedOver |
complex numbers
ⓘ
rational numbers ⓘ real numbers ⓘ |
| field |
Hodge theory
NERFINISHED
ⓘ
algebraic geometry ⓘ algebraic topology ⓘ complex geometry ⓘ |
| generalizes | pure Hodge structure ⓘ |
| hasComponent |
Hodge filtration
ⓘ
weight filtration ⓘ |
| hasFiltration |
decreasing Hodge filtration
ⓘ
increasing weight filtration ⓘ |
| hasInvariant |
Hodge numbers of graded pieces
ⓘ
weight filtration length ⓘ |
| hasMorphisms | strictly compatible with filtrations ⓘ |
| hasUnderlyingObject |
finite-dimensional complex vector space
ⓘ
finite-dimensional rational vector space ⓘ finite-dimensional real vector space ⓘ |
| HodgeFiltrationSymbol | F^• ⓘ |
| introducedBy | Pierre Deligne NERFINISHED ⓘ |
| introducedInContextOf | cohomology of algebraic varieties ⓘ |
| introducedInDecade | 1970s ⓘ |
| property |
each graded piece for the weight filtration carries a pure Hodge structure
ⓘ
functorial in cohomology ⓘ |
| relatedConcept |
Deligne cohomology
NERFINISHED
ⓘ
mixed Hodge module ⓘ variation of Hodge structure ⓘ weight filtration of monodromy ⓘ |
| satisfies |
Hodge decomposition on graded pieces
ⓘ
Hodge symmetry on graded pieces ⓘ |
| usedFor |
study of algebraic cycles
ⓘ
study of degenerations of Hodge structures ⓘ study of fundamental groups of varieties ⓘ study of motives ⓘ study of variation of mixed Hodge structure ⓘ |
| weightFiltrationSymbol | W_• ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.