Complex projective space

Construction

Complex projective space is a complex manifold that may be described by n + 1 complex coordinates as

${\displaystyle Z=(Z_1,Z_2,\dots ,Z_{n+1})\in {\mathbb{C}}^{n+1},(Z_1,Z_2,\dots ,Z_{n+1})\ne (0,0,\dots ,0)}$

where the tuples differing by an overall rescaling are identified:

${\displaystyle (Z_1,Z_2,\dots ,Z_{n+1})\equiv (\lambda Z_1,\lambda Z_2,\dots ,\lambda Z_{n+1});\lambda \in \mathbb{C},\lambda \ne 0.}$

That is, these are homogeneous coordinates in the traditional sense of projective geometry. The point set CPⁿ is covered by the patches ${\displaystyle U_i=\{Z\mid Z_i\ne 0\}}$. In U_i, one can define a coordinate system by

${\displaystyle z_1=Z_1/Z_i,z_2=Z_2/Z_i,\dots ,z_{i-1}=Z_{i-1}/Z_i,z_i=Z_{i+1}/Z_i,\dots ,z_n=Z_{n+1}/Z_i.}$

The coordinate transitions between two different such charts U_i and U_j are holomorphic functions (in fact they are fractional linear transformations). Thus CPⁿ carries the structure of a complex manifold of complex dimension n, and a fortiori the structure of a real differentiable manifold of real dimension 2n.

One may also regard CPⁿ as a quotient of the unit 2n + 1 sphere in Cⁿ⁺¹ under the action of U(1):

CPⁿ = S²ⁿ⁺¹/U(1).

This is because every line in Cⁿ⁺¹ intersects the unit sphere in a circle. By first projecting to the unit sphere and then identifying under the natural action of U(1) one obtains CPⁿ. For n = 1 this construction yields the classical Hopf bundle ${\displaystyle S^3\to S^2}$. From this perspective, the differentiable structure on CPⁿ is induced from that of S²ⁿ⁺¹, being the quotient of the latter by a compact group that acts properly.

Topology

The topology of CPⁿ is determined inductively by the following cell decomposition. Let H be a fixed hyperplane through the origin in Cⁿ⁺¹. Under the projection map Cⁿ⁺¹\{0} → CPⁿ, H goes into a subspace that is homeomorphic to CPⁿ⁻¹. The complement of the image of H in CPⁿ is homeomorphic to Cⁿ. Thus CPⁿ arises by attaching a 2n-cell to CPⁿ⁻¹:

${\displaystyle {\mathbf{C}\mathbf{P}}^n={\mathbf{C}\mathbf{P}}^{n-1}\cup {\mathbf{C}}^n.}$

Alternatively, if the 2n-cell is regarded instead as the open unit ball in Cⁿ, then the attaching map is the Hopf fibration of the boundary. An analogous inductive cell decomposition is true for all of the projective spaces; see (Besse 1978).

CW-decomposition

One useful way to construct the complex projective spaces ${\displaystyle {\mathbf{C}\mathbf{P}}^n}$ is through a recursive construction using CW-complexes. Recall that there is a homeomorphism ${\displaystyle {\mathbf{C}\mathbf{P}}^1\cong S^2}$ to the 2-sphere, giving the first space. We can then induct on the cells to get a pushout map $${\displaystyle \begin{array}{ccc}{S}^3& \hookrightarrow & D^4\\ \downarrow & & \downarrow \\ {\mathbf{C}\mathbf{P}}^1& \to & {\mathbf{C}\mathbf{P}}^2\end{array}}$$ where ${\displaystyle D^4}$ is the four ball, and ${\displaystyle S^3\to {\mathbf{C}\mathbf{P}}^1}$ represents the generator in ${\displaystyle {\pi }_3(S^2)}$ (hence it is homotopy equivalent to the Hopf map). We can then inductively construct the spaces as pushout diagrams $${\displaystyle \begin{array}{ccc}{S}^{2n-1}& \hookrightarrow & D^{2n}\\ \downarrow & & \downarrow \\ {\mathbf{C}\mathbf{P}}^{n-1}& \to & {\mathbf{C}\mathbf{P}}^n\end{array}}$$ where ${\displaystyle S^{2n-1}\to {\mathbf{C}\mathbf{P}}^{n-1}}$ represents an element in $${\displaystyle \begin{array}{cc}{\pi }_{2n-1}({\mathbf{C}\mathbf{P}}^{n-1})& \cong {\pi }_{2n-1}(S^{2n-2})\\ & \cong \mathbb{Z}/2\end{array}}$$ The isomorphism of homotopy groups is described below, and the isomorphism of homotopy groups is a standard calculation in stable homotopy theory (which can be done with the Serre spectral sequence, Freudenthal suspension theorem, and the Postnikov tower). The map comes from the fiber bundle $${\displaystyle S^1\hookrightarrow S^{2n-1}\twoheadrightarrow {\mathbf{C}\mathbf{P}}^{n-1}}$$ giving a non-contractible map, hence it represents the generator in ${\displaystyle \mathbb{Z}/2}$. Otherwise, there would be a homotopy equivalence ${\displaystyle {\mathbf{C}\mathbf{P}}^n\simeq {\mathbf{C}\mathbf{P}}^{n-1}\times D^n}$, but then it would be homotopy equivalent to ${\displaystyle S^2}$, a contradiction which can be seen by looking at the homotopy groups of the space.

Point-set topology

Complex projective space is compact and connected, being a quotient of a compact, connected space.

Homotopy groups

From the fiber bundle

${\displaystyle S^1\hookrightarrow S^{2n+1}\twoheadrightarrow {\mathbf{C}\mathbf{P}}^n}$

or more suggestively

${\displaystyle U(1)\hookrightarrow S^{2n+1}\twoheadrightarrow {\mathbf{C}\mathbf{P}}^n}$

CPⁿ is simply connected. Moreover, by the long exact homotopy sequence, the second homotopy group is π₂(CPⁿ) ≅ Z, and all the higher homotopy groups agree with those of S²ⁿ⁺¹: π_k(CPⁿ) ≅ π_k(S²ⁿ⁺¹) for all k > 2.

Homology

In general, the algebraic topology of CPⁿ is based on the rank of the homology groups being zero in odd dimensions; also H_2i(CPⁿ, Z) is infinite cyclic for i = 0 to n. Therefore, the Betti numbers run

1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ...

That is, 0 in odd dimensions, 1 in even dimensions 0 through 2n. The Euler characteristic of CPⁿ is therefore n + 1. By Poincaré duality the same is true for the ranks of the cohomology groups. In the case of cohomology, one can go further, and identify the graded ring structure, for cup product; the generator of H²(CPⁿ, Z) is the class associated to a hyperplane, and this is a ring generator, so that the ring is isomorphic with

Z[T]/(Tⁿ⁺¹),

with T a degree two generator. This implies also that the Hodge number h^i,i = 1, and all the others are zero. See (Besse 1978).

K-theory

It follows from induction and Bott periodicity that

${\displaystyle K_{\mathbf{C}}^{\ast }({\mathbf{C}\mathbf{P}}^n)=K_{\mathbf{C}}^0({\mathbf{C}\mathbf{P}}^n)=\mathbf{Z}[H]/(H-1{)}^{n+1}.}$

The tangent bundle satisfies

${\displaystyle T{\mathbf{C}\mathbf{P}}^n\oplus {\vartheta }^1=H^{\oplus n+1},}$

where ${\displaystyle {\vartheta }^1}$ denotes the trivial line bundle, from the Euler sequence. From this, the Chern classes and characteristic numbers can be calculated explicitly.

Classifying space

There is a space ${\displaystyle {\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}}$ which, in a sense, is the inductive limit of ${\displaystyle {\mathbf{C}\mathbf{P}}^n}$ as ${\displaystyle n\to \mathrm{\infty }}$. It is BU(1), the classifying space of U(1), the circle group, in the sense of homotopy theory, and so classifies complex line bundles. Equivalently it accounts for the first Chern class. This can be seen heuristically by looking at the fiber bundle maps $${\displaystyle S^1\hookrightarrow S^{2n+1}\twoheadrightarrow {\mathbf{C}\mathbf{P}}^n}$$ and ${\displaystyle n\to \mathrm{\infty }}$. This gives a fiber bundle (called the universal circle bundle) $${\displaystyle S^1\hookrightarrow S^{\mathrm{\infty }}\twoheadrightarrow {\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}}$$ constructing this space. Note using the long exact sequence of homotopy groups, we have ${\displaystyle {\pi }_2({\mathbf{C}\mathbf{P}}^{\mathrm{\infty }})={\pi }_1(S^1)}$ hence ${\displaystyle {\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}}$ is an Eilenberg–MacLane space, a ${\displaystyle K(\mathbb{Z},2)}$. Because of this fact, and Brown's representability theorem, we have the following isomorphism $${\displaystyle H^2(X;\mathbb{Z})\cong [X,{\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}]}$$ for any nice CW-complex ${\displaystyle X}$. Moreover, from the theory of Chern classes, every complex line bundle ${\displaystyle L\to X}$ can be represented as a pullback of the universal line bundle on ${\displaystyle {\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}}$, meaning there is a pullback square $${\displaystyle \begin{array}{ccc}L& \to & \mathcal{L}\\ \downarrow & & \downarrow \\ X& \to & {\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}\end{array}}$$ where ${\displaystyle \mathcal{L}\to {\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}}$ is the associated vector bundle of the principal ${\displaystyle U(1)}$-bundle ${\displaystyle S^{\mathrm{\infty }}\to {\mathbf{C}\mathbf{P}}^{\mathrm{\infty }}}$. See, for instance, (Bott & Tu 1982) and (Milnor & Stasheff 1974).

Differential geometry

The natural metric on CPⁿ is the Fubini–Study metric, and its holomorphic isometry group is the projective unitary group PU(n+1), where the stabilizer of a point is

${\displaystyle \mathrm{P}(1\times \mathrm{U}(n))\cong \mathrm{P}\mathrm{U}(n).}$

It is a Hermitian symmetric space (Kobayashi & Nomizu 1996), represented as a coset space

${\displaystyle U(n+1)/(U(1)\times U(n))\cong SU(n+1)/S(U(1)\times U(n)).}$

The geodesic symmetry at a point p is the unitary transformation that fixes p and is the negative identity on the orthogonal complement of the line represented by p.

Geodesics

Through any two points p, q in complex projective space, there passes a unique complex line (a CP¹). A great circle of this complex line that contains p and q is a geodesic for the Fubini–Study metric. In particular, all of the geodesics are closed (they are circles), and all have equal length. (This is always true of Riemannian globally symmetric spaces of rank 1.)

The cut locus of any point p is equal to a hyperplane CPⁿ⁻¹. This is also the set of fixed points of the geodesic symmetry at p (less p itself). See (Besse 1978).

Sectional curvature pinching

It has sectional curvature ranging from 1/4 to 1, and is the roundest manifold that is not a sphere (or covered by a sphere): by the 1/4-pinched sphere theorem, any complete, simply connected Riemannian manifold with curvature strictly between 1/4 and 1 is diffeomorphic to the sphere. Complex projective space shows that 1/4 is sharp. Conversely, if a complete simply connected Riemannian manifold has sectional curvatures in the closed interval [1/4,1], then it is either diffeomorphic to the sphere, or isometric to the complex projective space, the quaternionic projective space, or else the Cayley plane F₄/Spin(9); see (Brendle & Schoen 2008).

Spin structure

The odd-dimensional projective spaces can be given a spin structure, the even-dimensional ones cannot.

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)

Zariski topology

In algebraic geometry, complex projective space can be equipped with another topology known as the Zariski topology (Hartshorne 1977, §II.2). Let S = C[Z₀,...,Z_n] denote the commutative ring of polynomials in the (n+1) variables Z₀,...,Z_n. This ring is graded by the total degree of each polynomial:

${\displaystyle S=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{S}_n.}$

Define a subset of CPⁿ to be closed if it is the simultaneous solution set of a collection of homogeneous polynomials. Declaring the complements of the closed sets to be open, this defines a topology (the Zariski topology) on CPⁿ.

Structure as a scheme

Another construction of CPⁿ (and its Zariski topology) is possible. Let S₊ ⊂ S be the ideal spanned by the homogeneous polynomials of positive degree:

${\displaystyle \underset{n>0}{⨁}{S}_n.}$

Define Proj S to be the set of all homogeneous prime ideals in S that do not contain S₊. Call a subset of Proj S closed if it has the form

${\displaystyle V(I)=\{p\in \mathrm{Proj}S\mid p\supseteq I\}}$

for some ideal I in S. The complements of these closed sets define a topology on Proj S. The ring S, by localization at a prime ideal, determines a sheaf of local rings on Proj S. The space Proj S, together with its topology and sheaf of local rings, is a scheme. The subset of closed points of Proj S is homeomorphic to CPⁿ with its Zariski topology. Local sections of the sheaf are identified with the rational functions of total degree zero on CPⁿ.

Line bundles

All line bundles on complex projective space can be obtained by the following construction. A function f : Cⁿ⁺¹\{0} → C is called homogeneous of degree k if

${\displaystyle f(\lambda z)={\lambda }^{k}f(z)}$

for all λ ∈ C\{0} and z ∈ Cⁿ⁺¹\{0}. More generally, this definition makes sense in cones in Cⁿ⁺¹\{0}. A set V ⊂ Cⁿ⁺¹\{0} is called a cone if, whenever v ∈ V, then λv ∈ V for all λ ∈ C\{0}; that is, a subset is a cone if it contains the complex line through each of its points. If U ⊂ CPⁿ is an open set (in either the analytic topology or the Zariski topology), let V ⊂ Cⁿ⁺¹\{0} be the cone over U: the preimage of U under the projection Cⁿ⁺¹\{0} → CPⁿ. Finally, for each integer k, let O(k)(U) be the set of functions that are homogeneous of degree k in V. This defines a sheaf of sections of a certain line bundle, denoted by O(k).

In the special case k = −1, the bundle O(−1) is called the tautological line bundle. It is equivalently defined as the subbundle of the product

${\displaystyle {\mathbf{C}}^{n+1}\times {\mathbf{C}\mathbf{P}}^n\to {\mathbf{C}\mathbf{P}}^n}$

whose fiber over L ∈ CPⁿ is the set

${\displaystyle \{(x,L)\mid x\in L\}.}$

These line bundles can also be described in the language of divisors. Let H = CPⁿ⁻¹ be a given complex hyperplane in CPⁿ. The space of meromorphic functions on CPⁿ with at most a simple pole along H (and nowhere else) is a one-dimensional space, denoted by O(H), and called the hyperplane bundle. The dual bundle is denoted O(−H), and the k^th tensor power of O(H) is denoted by O(kH). This is the sheaf generated by holomorphic multiples of a meromorphic function with a pole of order k along H. It turns out that

${\displaystyle O(kH)\cong O(k).}$

Indeed, if L(z) = 0 is a linear defining function for H, then L^−k is a meromorphic section of O(k), and locally the other sections of O(k) are multiples of this section.

Since H¹(CPⁿ,Z) = 0, the line bundles on CPⁿ are classified up to isomorphism by their Chern classes, which are integers: they lie in H²(CPⁿ,Z) = Z. In fact, the first Chern classes of complex projective space are generated under Poincaré duality by the homology class associated to a hyperplane H. The line bundle O(kH) has Chern class k. Hence every holomorphic line bundle on CPⁿ is a tensor power of O(H) or O(−H). In other words, the Picard group of CPⁿ is generated as an abelian group by the hyperplane class [H] (Hartshorne 1977).