In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, complex projective space is the
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
with respect to the field of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. By analogy, whereas the points of a
real projective space label the lines through the origin of a real
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the points of a complex projective space label the ''
complex'' lines through the origin of a complex Euclidean space (see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
. The space is denoted variously as P(C
''n''+1), P
''n''(C) or CP
''n''. When , the complex projective space CP
1 is the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
, and when , CP
2 is the
complex projective plane (see there for a more elementary discussion).
Complex projective space was first introduced by as an instance of what was then known as the "geometry of position", a notion originally due to
Lazare Carnot
Lazare Nicolas Marguerite, Count Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician. He was known as the "Organizer of Victory" in the French Revolutionary Wars and Napoleonic Wars.
Education and early ...
, a kind of
synthetic geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass ...
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
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...
that the complex projective spaces were the most natural domains in which to consider the solutions of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
equations –
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. In modern times, both the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and geometry of complex projective space are well understood and closely related to that of the
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. Indeed, in a certain sense the (2''n''+1)-sphere can be regarded as a family of circles parametrized by CP
''n'': this is the
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz ...
. Complex projective space carries a (
Kähler)
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathe ...
, 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
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
. In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, complex projective space is the home of
projective varieties
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
, a well-behaved class of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. In topology, the complex projective space plays an important role as a
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
for complex
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
s: families of complex lines parametrized by another space. In this context, the infinite union of projective spaces (
direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
), denoted CP
∞, is the classifying space
K(Z,2). In quantum physics, the
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
associated to a
pure state of a quantum mechanical system is a
probability amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the qu ...
, 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.
Introduction
The notion of a projective plane arises out of the idea of perspection in geometry and art: that it is sometimes useful to include in the Euclidean plane an additional "imaginary" line that represents the horizon that an artist, painting the plane, might see. Following each direction from the origin, there is a different point on the horizon, so the horizon can be thought of as the set of all directions from the origin. The Euclidean plane, together with its horizon, is called the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
, and the horizon is sometimes called a
line at infinity. 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
In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned ...
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. Here, let R
''n''+1 denote the
real coordinate space
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
of ''n''+1 dimensions, and regard the landscape to be painted as a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
in this space. Suppose that the eye of the artist is the origin in R
''n''+1. Then along each line through his eye, there is a point of the landscape or a point on its horizon. Thus the real projective space is the space of lines through the origin in R
''n''+1. Without reference to coordinates, this is the space of lines through the origin in an (''n''+1)-dimensional real
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
.
To describe the complex projective space in an analogous manner requires a generalization of the idea of vector, line, and direction. Imagine that instead of standing in a real Euclidean space, the artist is standing in a complex Euclidean space C
''n''+1 (which has real dimension 2''n''+2) and the landscape is a ''complex'' hyperplane (of real dimension 2''n''). Unlike the case of real Euclidean space, in the complex case there are directions in which the artist can look which do not see the landscape (because it does not have high enough dimension). However, in a complex space, there is an additional "phase" associated with the directions through a point, and by adjusting this phase the artist can guarantee that he typically sees the landscape. The "horizon" is then the space of directions, but such that two directions are regarded as "the same" if they differ only by a phase. The complex projective space is then the landscape (C
''n'') with the horizon attached "at infinity". Just like the real case, the complex projective space is the space of directions through the origin of C
''n''+1, where two directions are regarded as the same if they differ by a phase.
Construction
Complex projective space is a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
that may be described by ''n'' + 1 complex coordinates as
:
where the tuples differing by an overall rescaling are identified:
:
That is, these are
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometr ...
in the traditional sense of
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
. The point set CP
''n'' is covered by the patches
. In ''U''
''i'', one can define a coordinate system by
:
The coordinate transitions between two different such charts ''U''
''i'' and ''U''
''j'' are
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
s (in fact they are
fractional linear transformation
In mathematics, a linear fractional transformation is, roughly speaking, a transformation of the form
:z \mapsto \frac ,
which has an inverse. The precise definition depends on the nature of , and . In other words, a linear fractional transfo ...
s). Thus CP
''n'' carries the structure of a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
of complex dimension ''n'', and ''
a fortiori'' the structure of a real
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
of real dimension 2''n''.
One may also regard CP
''n'' as a
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the unit 2''n'' + 1
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
in C
''n''+1 under the action of
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
:
:CP
''n'' = ''S''
2''n''+1/U(1).
This is because every line in C
''n''+1 intersects the unit sphere in a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
. By first projecting to the unit sphere and then identifying under the natural action of U(1) one obtains CP
''n''. For ''n'' = 1 this construction yields the classical
Hopf bundle
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hop ...
. From this perspective, the differentiable structure on CP
''n'' is induced from that of ''S''
2''n''+1, being the quotient of the latter by a compact group that acts properly.
Topology
The topology of CP
''n'' is determined inductively by the following
cell decomposition. Let ''H'' be a fixed hyperplane through the origin in C
''n''+1. Under the projection map , ''H'' goes into a subspace that is homeomorphic to CP
''n''−1. The complement of the image of ''H'' in CP
''n'' is homeomorphic to C
''n''. Thus CP
''n'' arises by attaching a 2''n''-cell to CP
''n''−1:
:
Alternatively, if the 2''n''-cell is regarded instead as the open unit ball in C
''n'', 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 .
CW-decomposition
One useful way to construct the complex projective spaces
is through a recursive construction using
CW-complexes. Recall that there is a homeomorphism
to the 2-sphere, giving the first space. We can then induct on the cells to get a
pushout map where
is the four ball, and
represents the generator in
(hence it is homotopy equivalent to the
Hopf map). We can then inductively construct the spaces as pushout diagrams
where
represents an element in
The isomorphism of homotopy groups is described below, and the isomorphism of homotopy groups is a standard calculation in
stable homotopy theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...
(which can be done with the
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homolo ...
,
Freudenthal suspension theorem
In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the b ...
, and the
Postnikov tower
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with t ...
). The map comes from the
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
giving a non-contractible map, hence it represents the generator in
. Otherwise, there would be a homotopy equivalence
, but then it would be homotopy equivalent to
, a contradiction which can be seen by looking at the homotopy groups of the space.
Point-set topology
Complex projective space is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
and
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
, being a quotient of a compact, connected space.
Homotopy groups
From the fiber bundle
:
or more suggestively
:
CP
''n'' is
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
. Moreover, by the
long exact homotopy sequence, the second homotopy group is , and all the higher homotopy groups agree with those of ''S''
2''n''+1: for all ''k'' > 2.
Homology
In general, the
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
of CP
''n'' is based on the rank of the
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s being zero in odd dimensions; also ''H''
2''i''(CP
''n'', 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 up to 2n. The
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of CP
''n'' is therefore ''n'' + 1. By
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
the same is true for the ranks of the
cohomology group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
s. In the case of cohomology, one can go further, and identify the
graded ring structure, for
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
; the generator of ''H''
2(CP
n, Z) is the class associated to a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
, and this is a ring generator, so that the ring is isomorphic with
:Z
'T''(''T''
''n''+1),
with ''T'' a degree two generator. This implies also that the
Hodge number
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every co ...
''h''
''i'',''i'' = 1, and all the others are zero. See .
''K''-theory
It follows from induction and
Bott periodicity
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comp ...
that
:
The
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
satisfies
:
where
denotes the trivial line bundle, from the
Euler sequence In mathematics, the Euler sequence is a particular exact sequence of sheaves on ''n''-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an (n+1)-fold sum of the dual of the Serre ...
. From this, the
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
es and
characteristic number
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
s can be calculated explicitly.
Classifying space
There is a space
which, in a sense, is the
inductive limit of
as
. It is
BU(1), the
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
of
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
, in the sense of
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
, and so classifies complex
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
s; equivalently it accounts for the first
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
. This can be seen heuristically by looking at the fiber bundle maps
and
. This gives a fiber bundle (called the
universal circle bundle)
constructing this space. Note using the long exact sequence of homotopy groups, we have
hence
is an
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
, a
. Because of this fact, and
Brown's representability theorem
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor ''F'' on the homotopy category ''Hotc'' of pointed connected CW complexes, to the category of sets Set, to be ...
, we have the following isomorphism