Dolbeault cohomology classes

E551971

cohomology class invariant in complex geometry mathematical object

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (2)

Label	Occurrences
Dolbeault cohomology	5
Dolbeault cohomology classes canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T5837346 — 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: Dolbeault cohomology classes
Context triple: [Lefschetz operator, appliesTo, Dolbeault cohomology classes]

A. Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry 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. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
D. Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
E. Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
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: Dolbeault cohomology classes
Target entity description: 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.

A. Differential Analysis on Complex Manifolds
"Differential Analysis on Complex Manifolds" is a foundational mathematical monograph that systematically develops the theory of differential and complex geometry 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. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
D. Cartan theorems A and B
Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
E. Singular Points of Complex Hypersurfaces
"Singular Points of Complex Hypersurfaces" is a foundational monograph in singularity theory that systematically studies the local and topological properties of singularities arising in complex algebraic and analytic hypersurfaces.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	cohomology class ⓘ invariant in complex geometry ⓘ mathematical object ⓘ
arisesFrom	Dolbeault complex of (p,•)-forms ⓘ
associatedOperator	Dolbeault operator ⓘ linked to: Dolbeault operator ∂ ∂̄-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 ⓘ 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 ⓘ 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 ⓘ

How these facts were elicited