The Riemann sphere, the one-dimensional complex projective space, i.e. the complex projective line.
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(Cn+1), Pn(C) or CPn. When n = 1, the complex projective space CP1 is the Riemann sphere, and when n = 2, CP2 is the complex projective plane (see there for a more elementary discussion).
Complex projective space was first introduced by von Staudt (1860) as an instance of what was then known as the "geometry of position", a notion originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. 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 carries a (Kähler) metric, called the Fubini–Study metric, in terms of which it is a Hermitian symmetric space of rank 1.
Complex projective space has many applications in both mathematics and quantum physics. In algebraic geometry, complex projective space is the home of projective varieties, a well-behaved class of algebraic varieties. In topology, the complex projective space plays an important role as a classifying space for complex line bundles: families of complex lines parametrized by another space. In this context, the infinite union of projective spaces (direct limit), denoted CP∞, is the classifying space K(Z,2). 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.
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complexprojectivespace is the projectivespace with respect to the field of complex numbers. By analogy, whereas the points of a real projective space...
point", which is subject to the axioms of projective geometry. For some such set of axioms, the projectivespaces that are defined have been shown to be...
{\displaystyle w_{1}} , which has degree 1. Complexprojectivespace Quaternionic projectivespace Lens space Real projective plane See the table of Don Davis for...
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space P n {\displaystyle \mathbb {P}...
the complexprojective plane, usually denoted P2(C) or CP2, is the two-dimensional complexprojectivespace. It is a complex manifold of complex dimension...
compared to elementary Euclidean geometry, projective geometry has a different setting, projectivespace, and a selective set of basic geometric concepts...
simplest complex manifolds. In projective geometry, the sphere is an example of a complexprojectivespace and can be thought of as the complexprojective line...
quantum mechanics, the projective Hilbert space or ray space P ( H ) {\displaystyle \mathbf {P} (H)} of a complex Hilbert space H {\displaystyle H} is...
as the complexprojective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projectivespace. Not all projective planes...
mathematics, quaternionic projectivespace is an extension of the ideas of real projectivespace and complexprojectivespace, to the case where coordinates...
otherwise. A projectivecomplex analytic variety is a subset X ⊆ C P n {\displaystyle X\subseteq \mathbb {CP} ^{n}} of complexprojectivespace that is, in...
if a CW-complex has no cells in consecutive dimensions, then all of its homology modules are free. For example, the complexprojectivespace C P n {\displaystyle...
are holomorphic Complexprojectivespace, a projectivespace with respect to the field of complex numbers Unitary space, a vector space with the addition...
isometry group of complexprojectivespace, just as the projective orthogonal group is the isometry group of real projectivespace. In terms of matrices...
analogues of real and complexprojectivespace. Therefore the classifying space BC2 is of the homotopy type of RP∞, the real projectivespace given by an infinite...
connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.) Second, the product of symmetric spaces is symmetric,...
sheaf cohomology groups on complexprojectivespace. The projectivespace in question is the twistor space, a geometrical space naturally associated to the...
{\displaystyle \mathbb {Z} [x]/x^{n+1}.} In the case of infinite complexprojectivespace, taking limits gives the answer Z [ x ] . {\displaystyle \mathbb...
fibers over real projectivespace RPn with fiber S0. The Hopf construction gives circle bundles p : S2n+1 → CPn over complexprojectivespace. This is actually...
particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic...
the complex vector space Cn+1{\displaystyle \mathbb {C} ^{n+1}}. The projective model of the complex hyperbolic space is the projectivized space of all...
Examples hyperbolic space Gauss–Bolyai–Lobachevsky space Grassmannian Complexprojectivespace Real projectivespace Euclidean space Stiefel manifold Upper...
Picard group of the projectivespace. In Michael Atiyah's "K-theory", the tautological line bundle over a complexprojectivespace is called the standard...
says that the geometry of projectivecomplex analytic spaces (or manifolds) is equivalent to the geometry of projectivecomplex varieties. The combination...
especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action...
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