Fubini–Study form
E551964
The Fubini–Study form is the canonical Kähler form on complex projective space, encoding its standard Hermitian and symplectic geometry.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
(1,1)-form
ⓘ
Kähler form ⓘ closed 2-form ⓘ real differential form ⓘ symplectic form ⓘ |
| arisesFrom | Chern connection on O(1) ⓘ |
| definedOn |
CP^1
NERFINISHED
ⓘ
CP^n NERFINISHED ⓘ CP^∞ ⓘ complex projective space NERFINISHED ⓘ |
| determines | standard volume form on CP^n ⓘ |
| givesComplexStructureCompatibilityTo | CP^n NERFINISHED ⓘ |
| givesSymplecticStructureTo | CP^n NERFINISHED ⓘ |
| hasAssociatedMetric | Fubini–Study metric NERFINISHED ⓘ |
| hasConstantHolomorphicSectionalCurvature | yes (for associated metric) ⓘ |
| induces | standard Riemannian metric on CP^n ⓘ |
| isAssociatedWithLineBundle | hyperplane line bundle O(1) ⓘ |
| isCanonicalOn |
CP^n
NERFINISHED
ⓘ
complex projective space ⓘ |
| isClosed | yes ⓘ |
| isCompatibleWith |
standard Hermitian structure on CP^n
ⓘ
standard complex structure on CP^n ⓘ |
| isExact | no ⓘ |
| isExampleOf |
Hodge form on a projective variety
ⓘ
Kähler form coming from an ample line bundle ⓘ |
| isHomogeneousUnder | action of U(n+1) on CP^n ⓘ |
| isInvariantUnder |
holomorphic isometries of CP^n
ⓘ
projective unitary group PU(n+1) NERFINISHED ⓘ unitary group U(n+1) NERFINISHED ⓘ |
| isKählerFormOf | Fubini–Study metric NERFINISHED ⓘ |
| isNamedAfter |
Eduard Study
NERFINISHED
ⓘ
Guido Fubini NERFINISHED ⓘ |
| isNormalizedSoThat |
integral over CP^1 equals π
ⓘ
integral over projective line equals 1 (up to conventions) ⓘ |
| isPositive | yes ⓘ |
| isPullbackOf | standard symplectic form on C^{n+1}\{0} via Hopf fibration (up to normalization) ⓘ |
| isUniqueUpToScaleAs | U(n+1)-invariant Kähler form on CP^n ⓘ |
| isUsedIn |
Kähler geometry
NERFINISHED
ⓘ
algebraic geometry ⓘ complex differential geometry ⓘ geometric quantization ⓘ moment map theory ⓘ study of projective embeddings ⓘ symplectic geometry ⓘ |
| locallyExpressedInAffineChart | i∂∂̄ log(1+∑_{j=1}^n |w_j|^2) ⓘ |
| locallyGivenBy | i∂∂̄ log(∑|z_i|^2) ⓘ |
| representsCohomologyClass |
first Chern class of O(1)
ⓘ
generator of H^2(CP^n, Z) ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.