Complex projective space

For $n\ge0$ , complex projective $n$ -space is the quotient

\mathbb{CP}^n=(\mathbb C^{n+1}\setminus\{0\})/\mathbb C^\times,

where $\mathbb C^\times$ acts by scalar multiplication and the quotient has the quotient topology. Equivalently, its points are one-dimensional complex linear subspaces of $\mathbb C^{n+1}$ . For each $i$ , the sets $U_i=\{[z_0:\cdots:z_n]:z_i\ne0\}$ are identified with $\mathbb C^n$ by the coordinate ratios $z_j/z_i$ for $j\ne i$ . On overlaps these ratios give holomorphic transition maps, so the charts define a complex manifold of complex dimension $n$ and real dimension $2n$ .

Comparison with algebraic geometry

This complex manifold is also the analytification associated with the scheme-theoretic projective space $\mathbb P^n_{\mathbb C}$ , via the usual complex points and their analytic structure.

Quotient and homogeneous-space descriptions

Every complex line meets the unit sphere in a circle, so there is also the Hopf quotient

\mathbb{CP}^n\cong S^{2n+1}/U(1).

The unitary group acts transitively on complex lines, with stabilizer $U(1)\times U(n)$ , yielding

\mathbb{CP}^n\cong U(n+1)/(U(1)\times U(n)).

These descriptions exhibit $\mathbb{CP}^n$ as a compact connected smooth homogeneous space.

Complex, symplectic, and Kähler structure

The Fubini–Study metric is invariant under the projective unitary action, and its fundamental two-form is closed. Consequently $\mathbb{CP}^n$ is a Kähler manifold. Its Kähler class lies on the positive ray through the canonical integral generator of $H^2(\mathbb{CP}^n;\mathbb Z)$ . With the normalization $\omega_{\mathrm{FS}}=i\partial\bar\partial\log(1+\lVert z\rVert^2)$ , the integral generator is $[\omega_{\mathrm{FS}}/(2\pi)]$ .

The group $\operatorname{PGL}_{n+1}(\mathbb C)$ acts transitively by holomorphic transformations, but it does not preserve a chosen Fubini–Study metric in general; the projective unitary subgroup does. Thus the holomorphic and isometric symmetry groups should not be conflated.

Important low-dimensional case

The complex projective line $\mathbb{CP}^1$ is the Riemann sphere. It is diffeomorphic to $S^2$ , while for higher $n$ , $\mathbb{CP}^n$ is not a sphere.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: Chapter 1, projective space as a complex manifold, and Chapter 3, the Fubini–Study Kähler form.
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Publisher record. Relevant: quotient manifolds and homogeneous spaces.