Complex projective plane
In mathematics, 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 complex projective space. It is a complex manifold 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.