Hodge theory

E129504

branch of mathematics theory in algebraic geometry theory in differential geometry

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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All labels observed (2)

Label	Occurrences
Hodge theory canonical	11
Hodge decomposition	1

Statements (49)

Predicate	Object
instanceOf	branch of mathematics ⓘ theory in algebraic geometry ⓘ theory in differential geometry ⓘ
appliesTo	compact Kähler manifolds ⓘ compact Riemannian manifolds ⓘ complex manifolds ⓘ complex projective varieties ⓘ
developedBy	W. V. D. Hodge ⓘ
fieldOfStudy	Kähler identities ⓘ surface form: Kähler geometry algebraic geometry ⓘ cohomology ⓘ complex geometry ⓘ differential forms ⓘ topology of manifolds ⓘ
hasSubfield	classical Hodge theory ⓘ mixed Hodge theory ⓘ non-abelian Hodge theory ⓘ p-adic Hodge theory ⓘ
influenced	mirror symmetry ⓘ modern algebraic geometry ⓘ string theory ⓘ
provides	Hodge decomposition of cohomology groups ⓘ constraints on Betti numbers ⓘ decomposition of cohomology into types (p,q) ⓘ isomorphism between de Rham cohomology and harmonic forms ⓘ symmetries of Hodge numbers ⓘ
relatedTo	Dolbeault cohomology classes ⓘ surface form: Dolbeault cohomology Lefschetz theory ⓘ Morse Theory ⓘ surface form: Morse theory de Rham cohomology ⓘ representation theory of Lie groups ⓘ singular cohomology ⓘ
studies	Dolbeault cohomology classes ⓘ surface form: Dolbeault cohomology Hodge decomposition ⓘ Hodge filtration ⓘ Hodge numbers ⓘ Levi-Civita symbol ⓘ surface form: Hodge star operator Hodge structures ⓘ Hodge–Riemann bilinear relations ⓘ Kähler identities ⓘ Laplacian on differential forms ⓘ harmonic differential forms ⓘ relationships between differential forms and cohomology classes ⓘ variation of Hodge structure ⓘ
uses	Kähler metrics ⓘ Riemannian metrics ⓘ complex structures ⓘ elliptic differential operators ⓘ functional analysis ⓘ

Referenced by (12)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kähler manifold → usedIn → Hodge theory ⓘ

Pierre Deligne → fieldOfWork → Hodge theory ⓘ

Kähler form → usedIn → Hodge theory ⓘ

Lefschetz operator → field → Hodge theory ⓘ

Lefschetz operator → relatedConcept → Hodge theory ⓘ

this entity surface form: Hodge decomposition

Poincaré lemma → relatedTo → Hodge theory ⓘ

Poincaré duality → relatedConcept → Hodge theory ⓘ

Hodge Conjecture → field → Hodge theory ⓘ

William Vallance Douglas Hodge → fieldOfWork → Hodge theory ⓘ

William Vallance Douglas Hodge → notableWork → Hodge theory ⓘ

Differential Analysis on Complex Manifolds → topic → Hodge theory ⓘ

Lefschetz hyperplane theorem → usedIn → Hodge theory ⓘ