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gptkbp:instanceOf
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gptkb:geometry
gptkb:mathematical_concept
Complex manifold
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gptkbp:appearsIn
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gptkb:Algebraic_topology
gptkb:string_theory
gptkb:Twistor_theory
Quantum mechanics (projective Hilbert space)
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gptkbp:automorphismGroup
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PGL(n+1,C)
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gptkbp:built
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Set of lines through the origin in C^{n+1}
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gptkbp:can_be_covered_by
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n+1 affine charts
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gptkbp:cohomology_ring
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Z[x]/(x^{n+1})
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gptkbp:complex_dimension
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n
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gptkbp:CW_complex_structure
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Yes
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gptkbp:definedIn
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gptkb:Complex_numbers
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gptkbp:dimensions
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n (for CP^n)
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gptkbp:Euler_characteristic
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n+1
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gptkbp:field
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gptkb:Mathematics
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gptkbp:first_Betti_number
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0
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gptkbp:first_Chern_class
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Generator of H^2(CP^n, Z)
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gptkbp:fundamentalGroup
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Trivial for n = 1
Trivial for n > 0
Trivial for n > 1
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gptkbp:has_cell_decomposition
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One cell in each even dimension up to 2n
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gptkbp:hasModel
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Projective geometry over C
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gptkbp:hasSpecialCase
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CP^1 is the Riemann sphere
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gptkbp:heldBy
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gptkb:algebraic_geometry
Kähler manifold
Smooth manifold
Compact complex manifold
Simply connected manifold for n > 0
Symplectic manifold
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gptkbp:homogeneous_space
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Yes
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gptkbp:homogeneous_space_under
|
Projective linear group PGL(n+1,C)
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https://www.w3.org/2000/01/rdf-schema#label
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Complex projective space
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gptkbp:notation
|
CP^n
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gptkbp:real_dimension
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2n
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gptkbp:relatedTo
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Grassmannian
Quaternionic projective space
Real projective space
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gptkbp:second_Betti_number
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1
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gptkbp:topology
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gptkb:Hausdorff
Connected
Compact
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gptkbp:used_in
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gptkb:Topology
gptkb:Algebraic_geometry
Mathematical physics
Complex geometry
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gptkbp:bfsParent
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gptkb:CW_Complexes
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gptkbp:bfsLayer
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8
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