Complex projective space
Subsequently, near the turn of the 20th century it became clear to the Italian school of algebraic geometry that the complex projective spaces were the most natural domains in which to consider the solutions of polynomial equations – algebraic varieties (Grattan-Guinness 2005, pp. 445–446).
In modern times, both the topology and geometry of complex projective space are well understood and closely related to that of the sphere.
Indeed, in a certain sense the (2n+1)-sphere can be regarded as a family of circles parametrized by CPn: this is the Hopf fibration.
Complex projective space has many applications in both mathematics and quantum physics.
In quantum physics, the wave function associated to a pure state of a quantum mechanical system is a probability amplitude, meaning that it has unit norm, and has an inessential overall phase: that is, the wave function of a pure state is naturally a point in the projective Hilbert space of the state space.
By the same construction, projective spaces can be considered in higher dimensions.
For instance, the real projective 3-space is a Euclidean space together with a plane at infinity that represents the horizon that an artist (who must, necessarily, live in four dimensions) would see.
These real projective spaces can be constructed in a slightly more rigorous way as follows.
To describe the complex projective space in an analogous manner requires a generalization of the idea of vector, line, and direction.
The complex projective space is then the landscape (Cn) with the horizon attached "at infinity".
In Ui, one can define a coordinate system by The coordinate transitions between two different such charts Ui and Uj 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.
By first projecting to the unit sphere and then identifying under the natural action of U(1) one obtains CPn.
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.
An analogous inductive cell decomposition is true for all of the projective spaces; see (Besse 1978).
From the fiber bundle or more suggestively CPn is simply connected.
It follows from induction and Bott periodicity that The tangent bundle satisfies where
denotes the trivial line bundle, from the Euler sequence.
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 It is a Hermitian symmetric space (Kobayashi & Nomizu 1996), represented as a coset space 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. Through any two points p, q in complex projective space, there passes a unique complex line (a CP1).
(This is always true of Riemannian globally symmetric spaces of rank 1.)
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).
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.
This ring is graded by the total degree of each polynomial: Define a subset of CPn to be closed if it is the simultaneous solution set of a collection of homogeneous polynomials.
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.
All line bundles on complex projective space can be obtained by the following construction.
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).
It is equivalently defined as the subbundle of the product whose fiber over L ∈ CPn is the set These line bundles can also be described in the language of divisors.
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).
