Statements (50)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
gptkb:Kähler_manifold gptkb: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
|
| 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:Type_IIB_string_theory
gptkb:6_String_Theory gptkb:Type_II_string_theory |
| gptkbp:bfsLayer |
7
|
| https://www.w3.org/2000/01/rdf-schema#label |
Calabi-Yau manifolds
|