Hodge–Riemann bilinear relations

E551973

mathematical theorem result in Hodge theory

The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.

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All labels observed (1)

Label	Occurrences
Hodge–Riemann bilinear relations canonical	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in Hodge theory ⓘ
appearsIn	classical Hodge theory of compact Kähler manifolds ⓘ modern treatments of algebraic geometry textbooks ⓘ
appliesTo	compact Kähler manifolds ⓘ smooth projective varieties over the complex numbers ⓘ
assumes	existence of a Kähler metric ⓘ finite-dimensional cohomology groups ⓘ
concerns	Hermitian form induced by Kähler class ⓘ bilinear form on cohomology groups ⓘ
describes	orthogonality properties of intersection forms ⓘ positivity properties of intersection forms ⓘ
field	Hodge theory NERFINISHED ⓘ algebraic geometry ⓘ complex geometry ⓘ
formalizes	positivity of the cup product with powers of a Kähler class ⓘ signature behavior of the intersection form on primitive subspaces ⓘ
generalizedBy	Hodge–Riemann relations for intersection cohomology NERFINISHED ⓘ Hodge–Riemann relations in combinatorial Hodge theory NERFINISHED ⓘ
gives	orthogonal decomposition of cohomology into primitive parts ⓘ sign constraints on intersection pairings ⓘ
historicalContext	developed in the 20th century ⓘ
holdsIn	cohomology with complex coefficients ⓘ middle-degree cohomology ⓘ
implies	Hodge index theorem NERFINISHED ⓘ hard Lefschetz theorem NERFINISHED ⓘ signature properties of intersection pairings ⓘ
motivated	Hodge–Riemann relations for polytopes and matroids NERFINISHED ⓘ generalizations to mixed Hodge structures ⓘ
namedAfter	Bernhard Riemann NERFINISHED ⓘ W. V. D. Hodge NERFINISHED ⓘ
property	definiteness of the intersection form on primitive classes ⓘ orthogonality of different primitive components ⓘ positivity on primitive cohomology ⓘ
relatedTo	Kähler identities ⓘ Lefschetz decomposition NERFINISHED ⓘ Weil conjectures NERFINISHED ⓘ
role	foundational tool in Kähler geometry ⓘ key ingredient in proofs of Lefschetz-type theorems ⓘ
usedIn	proofs of inequalities for intersection numbers ⓘ study of ample line bundles ⓘ study of the Kähler cone ⓘ study of the topology of algebraic varieties ⓘ
usesConcept	Hodge decomposition NERFINISHED ⓘ Lefschetz operator NERFINISHED ⓘ intersection form on cohomology ⓘ primitive cohomology ⓘ

Referenced by (1)

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

Hodge theory → studies → Hodge–Riemann bilinear relations ⓘ