Complex projective space is the space of complex lines through the origin in , that is the quotient where acts by scalar multiplication. A point is written in homogeneous coordinates , defined only up to a common nonzero scalar, so for all .
The standard affine charts cover , with coordinates for giving a biholomorphism . The transition maps are holomorphic, so is a compact complex manifold of dimension . The real analogue is a compact real manifold of dimension .
is the natural home for projective varieties. A homogeneous polynomial of degree satisfies , so although its value at a point is not well defined, its zero locus is, and this cuts out a well-defined subvariety. Compactness makes intersection theory well behaved and forces such loci to be closed. Hypersurfaces of degree therefore live here, for instance the quintic threefold sits in as the zero locus of a degree homogeneous polynomial in five variables.
The tautological line bundle assigns to each point the line it represents in , and its dual, the hyperplane bundle , has global sections given by linear forms . Its powers have global sections the degree homogeneous polynomials, which is why controls the projective geometry.