Complex projective plane

In mathematics, the complex projective plane, usually denoted ⁠${\displaystyle \mathbb {P} ^{2}(\mathbb {C} )}$⁠ or ⁠${\displaystyle \mathbb {CP} ^{2},}$⁠ is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates

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

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

${\displaystyle (Z_{1},Z_{2},Z_{3})\equiv (\lambda Z_{1},\lambda Z_{2},\lambda Z_{3});\quad \lambda \in \mathbb {C} ,\qquad \lambda \neq 0.}$

That is, these are homogeneous coordinates in the traditional sense of projective geometry.