Complex projective plane

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the complex projective plane, usually denoted ⁠
 
 
 
 
 
 P
 
 
 2
 
 
 (
 
 C
 
 )
 
 
 {\displaystyle \mathbb {P} ^{2}(\mathbb {C} )}
 
⁠ or ⁠
 
 
 
 
 
 C
 P
 
 
 2
 
 
 ,
 
 
 {\displaystyle \mathbb {CP} ^{2},}
 
⁠ is the two-dimensional <a href="/facts/Complex_projective_space/QmgneIzS">complex projective space</a>. It is a <a href="/facts/Complex_manifold/DlvPnpzm">complex manifold</a> of complex dimension 2, described by three complex coordinates

(
        
          Z
          
            1
          
        
        ,
        
          Z
          
            2
          
        
        ,
        
          Z
          
            3
          
        
        )
        ∈
        
          
            C
          
          
            3
          
        
        ,
        
        (
        
          Z
          
            1
          
        
        ,
        
          Z
          
            2
          
        
        ,
        
          Z
          
            3
          
        
        )
        ≠
        (
        0
        ,
        0
        ,
        0
        )
      
    
    {\displaystyle (Z_{1},Z_{2},Z_{3})\in \mathbb {C} ^{3},\qquad (Z_{1},Z_{2},Z_{3})\neq (0,0,0)}

where, however, the triples differing by an overall rescaling are identified:

(
        
          Z
          
            1
          
        
        ,
        
          Z
          
            2
          
        
        ,
        
          Z
          
            3
          
        
        )
        ≡
        (
        λ
        
          Z
          
            1
          
        
        ,
        λ
        
          Z
          
            2
          
        
        ,
        λ
        
          Z
          
            3
          
        
        )
        ;
        
        λ
        ∈
        
          C
        
        ,
        
        λ
        ≠
        0.
      
    
    {\displaystyle (Z_{1},Z_{2},Z_{3})\equiv (\lambda Z_{1},\lambda Z_{2},\lambda Z_{3});\quad \lambda \in \mathbb {C} ,\qquad \lambda \neq 0.}

That is, these are <a href="/facts/Homogeneous_coordinates/TG9ebKN4">homogeneous coordinates</a> in the traditional sense of <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a>.

Complex projective plane open-in-new

Complex projective plane