Calabi-Yau manifolds

GPTKB entity

Statements (48)
Predicate Object
gptkbp:instanceOf gptkb:mathematical_concept
Kähler manifold
Ricci-flat manifold
gptkbp:defines gptkb:Ricci-flat_Kähler_manifold
compact Kähler manifold with vanishing first Chern class
gptkbp:dimensions arbitrary (commonly 3 or higher)
gptkbp:field gptkb:algebraic_geometry
gptkb:theoretical_physics
differential geometry
gptkbp:hasConjecture gptkb:Calabi_conjecture
https://www.w3.org/2000/01/rdf-schema#label Calabi-Yau manifolds
gptkbp:importantFor gptkb:algebraic_geometry
gptkb:string_theory
gptkb:mirror_symmetry
gptkbp:namedAfter gptkb:Eugenio_Calabi
gptkb:Shing-Tung_Yau
gptkbp:notableExample gptkb:K3_surface
gptkb:quintic_threefold
complex torus
gptkbp:property holonomy group SU(n)
vanishing first Chern class
holonomy group is a subgroup of SU(n)
simply connected (in many cases)
can have complex dimension 1 (elliptic curve), 2 (K3 surface), 3 (quintic threefold), etc.
can be constructed as hypersurfaces in projective space
Hodge numbers are important invariants
admit Ricci-flat metric
admit covariantly constant holomorphic volume form
can have moduli spaces of Kähler structures
can have moduli spaces of complex structures
important in enumerative geometry
used in compactification in string theory
used in supersymmetry breaking
used in the study of Gromov-Witten invariants
mirror symmetry relates pairs of Calabi-Yau manifolds
gptkbp:provenBy gptkb:Shing-Tung_Yau
gptkbp:relatedTo gptkb:butter
gptkb:Fano_variety
gptkb:K3_surface
gptkb:mirror_manifolds
gptkbp:solvedBy gptkb:Calabi_conjecture
gptkbp:studiedBy gptkb:Eugenio_Calabi
gptkb:Shing-Tung_Yau
gptkbp:usedIn gptkb:string_theory
compactification of extra dimensions
gptkbp:yearOfProof 1977
gptkbp:bfsParent gptkb:string_theory
gptkbp:bfsLayer 5