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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (7)

Label	Occurrences
Calabi–Yau manifolds	6
Calabi–Yau manifold canonical	3
Calabi–Yau threefolds	2
Calabi–Yau compactifications	1
Calabi–Yau geometry	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	K3 surface ⓘ linked to: Calabi–Yau manifold 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	mirror Calabi–Yau manifold ⓘ linked to: 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