Kähler manifold

E23190

Riemannian manifold complex manifold geometric structure symplectic manifold

A Kähler manifold is a complex manifold equipped with a Hermitian metric whose associated symplectic form is closed, making it simultaneously a complex, Riemannian, and symplectic manifold in a compatible way.

Observed surface forms (1)

Surface form	Occurrences
Kähler geometry	1

Statements (48)

Predicate	Object
instanceOf	Riemannian manifold ⓘ complex manifold ⓘ geometric structure ⓘ symplectic manifold ⓘ
associatedForm	Kähler form ⓘ fundamental 2-form ⓘ
definitionCondition	J is orthogonal with respect to the Riemannian metric ⓘ Kähler form is d-closed ⓘ ∇J = 0 for Levi-Civita connection ∇ ⓘ
dimension	complex dimension n ⓘ even real dimension ⓘ
field	algebraic geometry ⓘ complex geometry ⓘ differential geometry ⓘ symplectic geometry ⓘ
hasCohomologyProperty	Betti numbers satisfy b_{2k+1} is even ⓘ admits Hodge decomposition ⓘ satisfies Hodge symmetry ⓘ satisfies hard Lefschetz theorem ⓘ
hasCurvatureProperty	Riemann curvature tensor has Kähler symmetries ⓘ
hasExample	Calabi–Yau manifold ⓘ Riemann surface with any Hermitian metric ⓘ complex projective space CP^n with Fubini–Study metric ⓘ complex tori with flat metric ⓘ smooth projective algebraic variety over C ⓘ
hasOperator	Dolbeault operators ∂ and ∂̄ ⓘ Lefschetz operator ⓘ
hasProperty	Kähler form is closed ⓘ Kähler identities hold between ∂, ∂̄, and Lefschetz operators ⓘ Laplace–Beltrami operator equals Hodge Laplacian on forms ⓘ Levi-Civita connection preserves complex structure ⓘ complex structure is integrable ⓘ holonomy group is contained in U(n) ⓘ metric is Hermitian with respect to complex structure ⓘ symplectic form is compatible with complex structure ⓘ
hasStructure	Hermitian metric ⓘ Riemannian metric ⓘ complex structure ⓘ symplectic form ⓘ
implies	underlying manifold is Riemannian ⓘ underlying manifold is complex ⓘ underlying manifold is symplectic ⓘ
localCoordinateDescription	Kähler form is i∂∂̄ of a real-valued potential function (locally) ⓘ metric is given by a Kähler potential ⓘ
namedAfter	Erich Kähler ⓘ
usedIn	Hodge theory ⓘ algebraic geometry ⓘ string theory ⓘ

Referenced by (2)

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

Ricci flow → appliedIn → Kähler manifold ⓘ

this entity surface form: Kähler geometry

Riemannian manifolds → hasVariant → Kähler manifold ⓘ

subject surface form: Riemannian manifold