Dolbeault cohomology classes
E551971
Dolbeault cohomology classes are equivalence classes of differential forms on a complex manifold defined using the ∂̄-operator, encoding the manifold’s complex-analytic and geometric structure.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Dolbeault cohomology | 3 |
| Dolbeault cohomology class | 0 |
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology class
ⓘ
invariant in complex geometry ⓘ mathematical object ⓘ |
| arisesFrom | Dolbeault complex of (p,•)-forms ⓘ |
| associatedOperator |
Dolbeault operator
NERFINISHED
ⓘ
∂̄-operator ⓘ |
| belongsTo | Dolbeault cohomology group H^{p,q}(X) ⓘ |
| builtFrom |
differential form of type (p,q)
ⓘ
∂̄-closed (p,q)-form ⓘ |
| captures |
obstructions to extending holomorphic objects
ⓘ
obstructions to solving ∂̄-equations ⓘ |
| computedBy | harmonic (p,q)-forms on compact Kähler manifolds ⓘ |
| context |
algebraic geometry
ⓘ
complex differential geometry ⓘ several complex variables ⓘ |
| definedOn | complex manifold ⓘ |
| definedUsing |
image of ∂̄ on (p,q−1)-forms
ⓘ
kernel of ∂̄ on (p,q)-forms ⓘ |
| dependsOn | complex structure of the manifold ⓘ |
| dimensionCalled | Hodge number h^{p,q} ⓘ |
| encodes |
complex-analytic structure of a manifold
ⓘ
geometric structure of a complex manifold ⓘ |
| equivalenceRelation | forms differing by a ∂̄-exact form ⓘ |
| finiteDimensionalOn | compact complex manifold ⓘ |
| functorialIn | holomorphic maps of complex manifolds ⓘ |
| generalizes | cohomology of holomorphic line bundles ⓘ |
| gradedBy | bidegree (p,q) ⓘ |
| groupStructure | abelian group ⓘ |
| hasRepresentative | smooth ∂̄-closed (p,q)-form ⓘ |
| invariantUnder | biholomorphic maps ⓘ |
| isomorphicTo | sheaf cohomology group H^q(X,Ω^p_X) on complex manifolds ⓘ |
| nonzeroClassCondition | form is ∂̄-closed but not ∂̄-exact ⓘ |
| quotientOf | space of ∂̄-closed (p,q)-forms by ∂̄-exact (p,q)-forms ⓘ |
| relatedTo |
Hodge decomposition
NERFINISHED
ⓘ
de Rham cohomology ⓘ sheaf cohomology of holomorphic forms ⓘ |
| stableUnder | small deformations of complex structure (in many settings) ⓘ |
| uniquenessUpTo | addition of a ∂̄-exact form ⓘ |
| usedIn |
Hodge theory
NERFINISHED
ⓘ
classification of complex manifolds ⓘ deformation theory of complex structures ⓘ study of holomorphic vector bundles ⓘ |
| usedToDefine | Chern classes via curvature forms ⓘ |
| vanishesIf | class contains a ∂̄-exact representative ⓘ |
| zeroClassCondition | form is ∂̄-exact ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Dolbeault cohomology
this entity surface form:
Dolbeault cohomology
this entity surface form:
Dolbeault cohomology