Kähler geometry

E888039

branch of differential geometry

Kähler geometry is a branch of differential geometry studying complex manifolds equipped with a compatible symplectic form and Riemannian metric, leading to rich interactions between complex, symplectic, and Riemannian geometry.

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Observed surface forms (1)

Surface form	Occurrences
Kähler metrics	1

Statements (50)

Predicate	Object
instanceOf	branch of differential geometry ⓘ
appliesTo	Hermitian symmetric spaces ⓘ Riemann surfaces NERFINISHED ⓘ complex tori ⓘ projective manifolds ⓘ
centralObject	Kähler form ⓘ Kähler manifold ⓘ Kähler metric ⓘ
developedInPeriod	20th century ⓘ
fieldOfStudy	Riemannian geometry NERFINISHED ⓘ complex manifolds ⓘ symplectic geometry ⓘ
hasApplicationIn	gauge theory ⓘ mirror symmetry ⓘ moduli spaces of complex structures ⓘ string theory NERFINISHED ⓘ
hasKeyProperty	Hermitian metric with closed associated 2-form ⓘ Levi-Civita connection equals Chern connection NERFINISHED ⓘ closed Kähler form ⓘ holonomy contained in U(n) ⓘ parallel complex structure ⓘ
hasTheorem	Hard Lefschetz theorem NERFINISHED ⓘ Hodge decomposition theorem for Kähler manifolds NERFINISHED ⓘ Kodaira embedding theorem NERFINISHED ⓘ Kähler identities NERFINISHED ⓘ Lefschetz decomposition NERFINISHED ⓘ Yau's solution of the Calabi conjecture NERFINISHED ⓘ ∂∂̄-lemma on Kähler manifolds ⓘ
hasTool	Kähler cone ⓘ Kähler potential ⓘ Monge–Ampère equations NERFINISHED ⓘ moment map ⓘ
namedAfter	Erich Kähler NERFINISHED ⓘ
relatesTo	Calabi–Yau manifolds NERFINISHED ⓘ Dolbeault cohomology NERFINISHED ⓘ Einstein metrics ⓘ Hodge decomposition NERFINISHED ⓘ Hodge theory NERFINISHED ⓘ Kähler–Einstein metrics NERFINISHED ⓘ Ricci-flat metrics ⓘ algebraic geometry ⓘ complex algebraic varieties ⓘ de Rham cohomology NERFINISHED ⓘ
requiresCompatibilityCondition	Riemannian metric ⓘ complex structure ⓘ symplectic form ⓘ
studies	Kähler manifolds NERFINISHED ⓘ
usesConcept	Riemannian metric ⓘ complex structure ⓘ symplectic form ⓘ

Referenced by (5)

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

Kähler–Ricci flow → field → Kähler geometry ⓘ

Monge–Ampère equation → usedIn → Kähler geometry ⓘ

Erich Kähler → knownFor → Kähler geometry ⓘ

this entity surface form: Kähler metrics

Shing-Tung Yau → hasResearchInterest → Kähler geometry ⓘ