Kähler cone
E551967
The Kähler cone is the convex cone in the cohomology of a complex manifold consisting of classes that can be represented by Kähler forms, encoding its possible Kähler metrics and playing a central role in complex and algebraic geometry.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
convex cone
ⓘ
geometric object ⓘ subset of cohomology ⓘ |
| appearsIn |
Hodge theory
ⓘ
Yau's solution of the Calabi conjecture (as space of Kähler classes) ⓘ global Torelli theorems for K3 surfaces NERFINISHED ⓘ |
| characterizedBy |
classes representable by closed real (1,1)-forms
ⓘ
classes representable by positive definite (1,1)-forms ⓘ |
| consistsOf | cohomology classes of Kähler forms ⓘ |
| definedOn | complex manifold ⓘ |
| dependsOn |
Hodge decomposition of the manifold
ⓘ
complex structure of the manifold ⓘ |
| encodes | possible Kähler metrics on a complex manifold ⓘ |
| field |
algebraic geometry
ⓘ
complex geometry ⓘ |
| hasBoundaryDescribedBy |
classes of nef but not Kähler divisors (projective case)
ⓘ
degeneration of Kähler metrics ⓘ |
| invariantUnder | biholomorphisms ⓘ |
| is | set of cohomology classes [ω] with ω a Kähler form ⓘ |
| livesIn |
H^{1,1}(X,ℝ)
ⓘ
real (1,1)-cohomology ⓘ |
| mayBeTrivialIf | manifold is non-Kähler ⓘ |
| nonEmptyIf | manifold is Kähler ⓘ |
| property |
convex
ⓘ
open cone ⓘ salient cone ⓘ |
| relatedConcept |
Mori cone
ⓘ
ample cone ⓘ nef cone ⓘ pseudo-effective cone ⓘ |
| relatedTo |
Kähler class
ⓘ
Kähler form ⓘ Kähler metric ⓘ |
| structure | open convex cone in a finite-dimensional real vector space ⓘ |
| studiedBy |
Claire Voisin
NERFINISHED
ⓘ
Jean-Pierre Demailly NERFINISHED ⓘ Shigefumi Mori NERFINISHED ⓘ Vladimir Shokurov NERFINISHED ⓘ |
| subsetOf |
nef cone (for projective manifolds, interior)
ⓘ
positive cone ⓘ |
| usedIn |
Calabi–Yau geometry
NERFINISHED
ⓘ
birational geometry of projective varieties ⓘ classification of compact Kähler manifolds ⓘ minimal model program (via nef and movable cones) ⓘ mirror symmetry ⓘ study of moduli of Kähler metrics ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.