Calabi–Yau manifold

E129502

Kähler manifold Ricci-flat manifold Riemannian manifold algebraic variety complex manifold geometric object projective variety

A Calabi–Yau manifold is a special type of complex manifold with vanishing first Chern class that plays a central role in string theory compactifications and complex algebraic geometry.

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All labels observed (5)

Label	Occurrences
Calabi–Yau compactifications	1
Calabi–Yau manifold canonical	1
Calabi–Yau manifolds	1
K3 surface	1
mirror Calabi–Yau manifold	1

How this entity was disambiguated

This entity first appeared as the object of triple T1138349 — 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: Calabi–Yau manifold
Context triple: [Kähler manifold, hasExample, Calabi–Yau manifold]

A. Kähler 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.
B. Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
C. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
D. Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
E. Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
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: Calabi–Yau manifold
Target entity description: A Calabi–Yau manifold is a special type of complex manifold with vanishing first Chern class that plays a central role in string theory compactifications and complex algebraic geometry.

A. Kähler 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.
B. Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
C. Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
D. Riemann surfaces
Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
E. Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
F. None of above. chosen

Statements (63)

Predicate	Object
instanceOf	Kähler manifold ⓘ Ricci-flat manifold ⓘ Riemannian manifold ⓘ algebraic variety ⓘ complex manifold ⓘ geometric object ⓘ projective variety ⓘ
appearsIn	heterotic string compactifications ⓘ superstring compactification from 10D to 4D ⓘ type II string compactifications ⓘ
centralConceptIn	Strominger–Yau–Zaslow conjecture ⓘ mirror symmetry conjecture ⓘ string phenomenology ⓘ
developedBy	Eugenio Calabi ⓘ Shing-Tung Yau ⓘ
dimension	complex dimension n ≥ 1 ⓘ
example	Calabi–Yau manifold self-linksurface differs ⓘ surface form: K3 surface complete intersection Calabi–Yau threefold ⓘ complex torus with trivial canonical bundle and appropriate holonomy ⓘ quintic threefold in ℙ^4 ⓘ
fieldOfStudy	algebraic geometry ⓘ complex geometry ⓘ differential geometry ⓘ string theory ⓘ
hasInvariant	Euler characteristic ⓘ Hodge numbers ⓘ Kähler cone ⓘ Kähler moduli ⓘ Picard number ⓘ complex structure moduli ⓘ fundamental group ⓘ
hasProperty	Kähler form is closed ⓘ admits a Ricci-flat Kähler metric ⓘ admits a nowhere-vanishing holomorphic volume form ⓘ c1 = 0 in H^2(M,ℝ) ⓘ c1(M)=0 in H^2(M,ℤ) for algebraic Calabi–Yau ⓘ holonomy contained in SU(n) ⓘ vanishing first Chern class ⓘ
hasStructure	Calabi–Yau metric ⓘ covariantly constant spinor ⓘ holomorphic tangent bundle ⓘ trivial canonical bundle ⓘ
implies	Ricci curvature tensor vanishes ⓘ first Betti number b1 = 0 for simply connected case ⓘ preservation of some supersymmetry in compactifications ⓘ
mirrorTo	Calabi–Yau manifold self-linksurface differs ⓘ surface form: mirror Calabi–Yau manifold
namedAfter	Eugenio Calabi ⓘ Shing-Tung Yau ⓘ
relatedConcept	G2 manifold ⓘ Kähler–Einstein metric ⓘ canonical bundle ⓘ holonomy group SU(n) ⓘ special holonomy ⓘ
typicalDimension	complex dimension 3 in string theory ⓘ
usedIn	complex algebraic geometry ⓘ mathematical physics ⓘ mirror symmetry ⓘ string compactification ⓘ string theory ⓘ superstring theory ⓘ supersymmetric field theories ⓘ topological string theory ⓘ
YauTheorem	existence of Ricci-flat Kähler metric given c1=0 ⓘ

How these facts were elicited

Referenced by (5)

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

Kähler manifold → hasExample → Calabi–Yau manifold ⓘ

Calabi–Yau manifold → example → Calabi–Yau manifold self-linksurface differs ⓘ

this entity surface form: K3 surface

Calabi–Yau manifold → mirrorTo → Calabi–Yau manifold self-linksurface differs ⓘ

this entity surface form: mirror Calabi–Yau manifold

Seiberg–Witten theory → relatedTo → Calabi–Yau manifold ⓘ

this entity surface form: Calabi–Yau compactifications

Shing-Tung Yau → knownFor → Calabi–Yau manifold ⓘ

this entity surface form: Calabi–Yau manifolds

All labels observed (5)

How this entity was disambiguated Show

Statements (63)

How these facts were elicited Show

Referenced by (5)

How this entity was disambiguated

How these facts were elicited