Complex projective space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Complex_projective_space


00:00:52 1 Introduction
00:03:58 2 Construction
00:07:50 3 Topology
00:14:25 3.1 Point-set topology
00:15:31 3.2 Homotopy groups
00:17:19 3.3 Homology
00:17:39 3.4 iK/i-theory
00:19:31 3.5 Classifying space
00:21:05 4 Differential geometry
00:21:14 4.1 Geodesics
00:23:31 4.2 Sectional curvature pinching
00:24:20 5 Algebraic geometry
00:25:35 5.1 Zariski topology
00:26:37 5.2 Structure as a scheme
00:27:45 5.3 Line bundles
00:28:37 6 See also
00:30:12 7 References
00:32:19 Line bundles
00:37:00 0, the line bundles on CPn are classified up to isomorphism by their Chern classes, which are integers: they lie in H2(CPn,Z)
00:38:06 See also



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SUMMARY
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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.