Statements (139)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
Kähler manifold symplectic manifold Ricci-flat manifold differential manifold |
| gptkbp:alternativeName |
Calabi-Yau_manifold
Calabi–Yau_manifold Kähler_metric complex_manifold |
| gptkbp:appearsIn |
gptkb:string_theory
gptkb:M-theory |
| gptkbp:category |
gptkb:algebraic_geometry
differential geometry mathematical physics complex geometry |
| gptkbp:compact |
can be compact or non-compact
|
| gptkbp:conjectureProvedBy |
gptkb:Shing-Tung_Yau
|
| gptkbp:definedIn |
Kähler manifold
|
| gptkbp:dimensions |
arbitrary (commonly 3)
can be 1 (elliptic curve) can be 2 (K3 surface) commonly 3 (Calabi-Yau threefold) even (real dimension) arbitrary (commonly 3 or higher) |
| gptkbp:example |
gptkb:K3_surface
gptkb:Riemannian_manifold gptkb:quintic_threefold Kähler manifold complex projective space complex torus |
| gptkbp:field |
gptkb:algebraic_geometry
differential geometry complex geometry |
| gptkbp:firstChernClass |
zero
|
| gptkbp:firstDescribed |
1930s
|
| gptkbp:firstProved |
1977
|
| gptkbp:hasApplication |
compactification in string theory
supersymmetry preservation |
| gptkbp:hasComponent |
gptkb:Kähler_form
complex structure symplectic form |
| gptkbp:hasHolonomy |
gptkb:SU(n)
|
| gptkbp:hasProperty |
gptkb:Betti_numbers
gptkb:Kähler_form gptkb:Ricci-flat_metric gptkb:Calabi_conjecture Kähler manifold complex structure Hodge numbers admits a Ricci-flat Kähler metric holonomy group SU(n) simply connected (for n>1) vanishing first Chern class Einstein manifold Hermitian metric whose imaginary part is closed Levi-Civita connection preserves complex structure Ricci curvature can be defined admits Bochner technique admits Calabi conjecture admits Dolbeault cohomology admits Hodge decomposition admits Hodge symmetry admits Kähler class admits Kähler cone admits Kähler form admits Kähler identities admits Kähler metric admits Kähler potential admits Kähler–Einstein metrics admits Kähler–Ricci flow admits Lefschetz decomposition admits Yau's theorem admits hard Lefschetz theorem closed Kähler form holonomy group is contained in U(n) simply connected (in many cases) compact or non-compact complex dimension n holomorphic volume form holonomy group contained in SU(n) no global holomorphic 1-forms (for n>1) trivial canonical bundle used in F-theory used in M-theory used in superstring theory |
| gptkbp:hasSpecialCase |
Kähler manifold
symplectic manifold Hermitian manifold |
| gptkbp:heldBy |
gptkb:Riemannian_manifold
Hermitian metric compatible with complex structure compatible with symplectic structure |
| gptkbp:implies |
Kähler manifold
|
| gptkbp:introducedIn |
1932
|
| gptkbp:namedAfter |
gptkb:Erich_Kähler
gptkb:Eugenio_Calabi gptkb:Shing-Tung_Yau |
| gptkbp:provenBy |
gptkb:Shing-Tung_Yau
|
| gptkbp:relatedTo |
gptkb:butter
gptkb:Hodge_theory gptkb:Fano_variety gptkb:K3_surface gptkb:Batyrev_mirror_construction gptkb:Donaldson-Thomas_invariants gptkb:Gromov-Witten_invariants gptkb:Strominger-Yau-Zaslow_conjecture gptkb:Yau's_proof_of_Calabi_conjecture gptkb:Fubini–Study_metric gptkb:Ricci-flat_metric gptkb:D-branes gptkb:mirror_symmetry gptkb:topological_string_theory gptkb:mirror_manifold Kähler manifold moduli space flux compactification string compactification string landscape supersymmetry breaking Hodge numbers |
| gptkbp:satisfies |
Kähler condition
dω = 0 for Kähler form ω |
| gptkbp:structure |
gptkb:Riemannian_manifold
complex structure symplectic form |
| gptkbp:studiedBy |
gptkb:Eugenio_Calabi
gptkb:Shing-Tung_Yau |
| gptkbp:studiedIn |
gptkb:algebraic_geometry
differential geometry complex geometry |
| gptkbp:usedIn |
gptkb:algebraic_geometry
gptkb:theoretical_physics gptkb:Hodge_theory gptkb:string_theory gptkb:mirror_symmetry complex algebraic geometry moduli spaces |
| gptkbp:bfsParent |
gptkb:Shing-Tung_Yau
|
| gptkbp:bfsLayer |
4
|