Algebraic Geometry
Complex projective space is a special case of a Grassmannian, and is a homogeneous space for various Lie groups. It is a Kähler manifold carrying the Fubini-Study metric, which is essentially determined by symmetry properties. It also plays a central role in algebraic geometry; by Chow's theorem, any compact complex submanifold of CPn is the zero locus of a finite number of polynomials, and is thus a projective algebraic variety. See (Griffiths & Harris 1994)
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