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Complex projective space

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).

The Riemann sphere, the one-dimensional complex projective space, i.e. the complex projective line.

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.

Introduction

 
Parallel lines in the plane intersect at the vanishing point in the line at infinity.

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, 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 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 Rn+1 denote the real coordinate space of n+1 dimensions, and regard the landscape to be painted as a hyperplane in this space. Suppose that the eye of the artist is the origin in Rn+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 Rn+1. Without reference to coordinates, this is the space of lines through the origin in an (n+1)-dimensional real vector space.

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 Cn+1 (which has real dimension 2n+2) and the landscape is a complex hyperplane (of real dimension 2n). 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 (Cn) with the horizon attached "at infinity". Just like the real case, the complex projective space is the space of directions through the origin of Cn+1, where two directions are regarded as the same if they differ by a phase.

Construction

Complex projective space is a complex manifold 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 the traditional sense of projective geometry. The point set CPn is covered by the patches  . 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.

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  . 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:

 

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   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 (which can be done with the Serre spectral sequence, Freudenthal suspension theorem, and the Postnikov tower). The map comes from the fiber bundle
 
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 and connected, being a quotient of a compact, connected space.

Homotopy groups

From the fiber bundle

 

or more suggestively

 

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: πk(CPn) ≅ πk(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 H2i(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 up to 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 hi,i = 1, and all the others are zero. See (Besse 1978).

K-theory

It follows from induction and Bott periodicity that

 

The tangent bundle satisfies

 

where   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   which, in a sense, is the inductive limit of   as  . 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

 
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, a  . Because of this fact, and Brown's representability theorem, we have the following isomorphism
 
for any nice CW-complex  . Moreover, from the theory of Chern classes, every complex line bundle   can be represented as a pullback of the universal line bundle on  , meaning there is a pullback square
 
where   is the associated vector bundle of the principal  -bundle  . 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

 

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.

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,...,Zn] denote the commutative ring of polynomials in the (n+1) variables Z0,...,Zn. 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. 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:

 

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

 

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

 

for all λ ∈ C\{0} and zCn+1\{0}. More generally, this definition makes sense in cones in Cn+1\{0}. A set VCn+1\{0} is called a cone if, whenever vV, then λvV for all λ ∈ C\{0}; that is, a subset is a cone if it contains the complex line through each of its points. If UCPn is an open set (in either the analytic topology or the Zariski topology), let VCn+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

 

whose fiber over LCPn is the set

 

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 kth 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

 

Indeed, if L(z) = 0 is a linear defining function for H, then Lk 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).

See also

References

  • Besse, Arthur L. (1978), Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Berlin, New York: Springer-Verlag, ISBN 978-3-540-08158-6.
  • Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3.
  • Brendle, Simon; Schoen, Richard (2008), "Classification of manifolds with weakly 1/4-pinched curvatures", Acta Mathematica, 200: 1–13, arXiv:0705.3963, doi:10.1007/s11511-008-0022-7.
  • Grattan-Guinness, Ivor (2005), Landmark writings in western mathematics 1640–1940, Elsevier, ISBN 978-0-444-50871-3.
  • Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523.
  • Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
  • Klingenberg, Wilhelm (1982), Riemannian geometry, Walter de Greuter, ISBN 978-3-11-008673-7.
  • Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of Differential Geometry, Volume II, Wiley Classics Library edition, ISBN 978-0-471-15732-8.
  • Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Princeton University Press, MR 0440554.
  • von Staudt, Karl Georg Christian (1860), Beiträge zur Geometrie der Lage, Nuremberg.

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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 The Riemann sphere the one dimensional complex projective space i e the complex projective line 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 Kahler 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 Contents 1 Introduction 2 Construction 3 Topology 3 1 CW decomposition 3 2 Point set topology 3 3 Homotopy groups 3 4 Homology 3 5 K theory 3 6 Classifying space 4 Differential geometry 4 1 Geodesics 4 2 Sectional curvature pinching 4 3 Spin structure 5 Algebraic geometry 5 1 Zariski topology 5 2 Structure as a scheme 5 3 Line bundles 6 See also 7 ReferencesIntroduction Edit Parallel lines in the plane intersect at the vanishing point in the line at infinity 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 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 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 Rn 1 denote the real coordinate space of n 1 dimensions and regard the landscape to be painted as a hyperplane in this space Suppose that the eye of the artist is the origin in Rn 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 Rn 1 Without reference to coordinates this is the space of lines through the origin in an n 1 dimensional real vector space 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 Cn 1 which has real dimension 2n 2 and the landscape is a complex hyperplane of real dimension 2n 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 Cn with the horizon attached at infinity Just like the real case the complex projective space is the space of directions through the origin of Cn 1 where two directions are regarded as the same if they differ by a phase Construction EditComplex projective space is a complex manifold that may be described by n 1 complex coordinates as Z Z 1 Z 2 Z n 1 C n 1 Z 1 Z 2 Z n 1 0 0 0 displaystyle Z Z 1 Z 2 ldots Z n 1 in mathbb C n 1 qquad Z 1 Z 2 ldots Z n 1 neq 0 0 ldots 0 where the tuples differing by an overall rescaling are identified Z 1 Z 2 Z n 1 l Z 1 l Z 2 l Z n 1 l C l 0 displaystyle Z 1 Z 2 ldots Z n 1 equiv lambda Z 1 lambda Z 2 ldots lambda Z n 1 quad lambda in mathbb C qquad lambda neq 0 That is these are homogeneous coordinates in the traditional sense of projective geometry The point set CPn is covered by the patches U i Z Z i 0 displaystyle U i Z mid Z i neq 0 In Ui one can define a coordinate system by z 1 Z 1 Z i z 2 Z 2 Z i z i 1 Z i 1 Z i z i Z i 1 Z i z n Z n 1 Z i displaystyle z 1 Z 1 Z i quad z 2 Z 2 Z i quad dots quad z i 1 Z i 1 Z i quad z i Z i 1 Z i quad dots quad z n Z n 1 Z i 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 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 S 3 S 2 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 EditThe 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 C P n C P n 1 C n displaystyle mathbf CP n mathbf CP 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 Edit One useful way to construct the complex projective spaces C P n displaystyle mathbf CP n is through a recursive construction using CW complexes Recall that there is a homeomorphism C P 1 S 2 displaystyle mathbf CP 1 cong S 2 to the 2 sphere giving the first space We can then induct on the cells to get a pushout mapS 3 D 4 C P 1 C P 2 displaystyle begin matrix S 3 amp hookrightarrow amp D 4 downarrow amp amp downarrow mathbf CP 1 amp to amp mathbf CP 2 end matrix where D 4 displaystyle D 4 is the four ball and S 3 C P 1 displaystyle S 3 to mathbf CP 1 represents the generator in p 3 S 2 displaystyle pi 3 S 2 hence it is homotopy equivalent to the Hopf map We can then inductively construct the spaces as pushout diagrams S 2 n 1 D 2 n C P n 1 C P n displaystyle begin matrix S 2n 1 amp hookrightarrow amp D 2n downarrow amp amp downarrow mathbf CP n 1 amp to amp mathbf CP n end matrix where S 2 n 1 C P n 1 displaystyle S 2n 1 to mathbf CP n 1 represents an element in p 2 n 1 C P n 1 p 2 n 1 S 2 n 2 Z 2 displaystyle begin aligned pi 2n 1 mathbf CP n 1 amp cong pi 2n 1 S 2n 2 amp cong mathbb Z 2 end aligned 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 S 1 S 2 n 1 C P n 1 displaystyle S 1 hookrightarrow S 2n 1 twoheadrightarrow mathbf CP n 1 giving a non contractible map hence it represents the generator in Z 2 displaystyle mathbb Z 2 Otherwise there would be a homotopy equivalence C P n C P n 1 D n displaystyle mathbf CP n simeq mathbf CP n 1 times D n but then it would be homotopy equivalent to S 2 displaystyle S 2 a contradiction which can be seen by looking at the homotopy groups of the space Point set topology Edit Complex projective space is compact and connected being a quotient of a compact connected space Homotopy groups Edit From the fiber bundle S 1 S 2 n 1 C P n displaystyle S 1 hookrightarrow S 2n 1 twoheadrightarrow mathbf CP n or more suggestively U 1 S 2 n 1 C P n displaystyle U 1 hookrightarrow S 2n 1 twoheadrightarrow mathbf CP n CPn is simply connected Moreover by the long exact homotopy sequence the second homotopy group is p2 CPn Z and all the higher homotopy groups agree with those of S2n 1 pk CPn pk S2n 1 for all k gt 2 Homology Edit In general the algebraic topology of CPn is based on the rank of the homology groups being zero in odd dimensions also H2i 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 up to 2n The Euler characteristic of CPn is therefore n 1 By Poincare 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 hi i 1 and all the others are zero See Besse 1978 K theory Edit It follows from induction and Bott periodicity that K C C P n K C 0 C P n Z H H 1 n 1 displaystyle K mathbf C mathbf CP n K mathbf C 0 mathbf CP n mathbf Z H H 1 n 1 The tangent bundle satisfies T C P n ϑ 1 H n 1 displaystyle T mathbf CP n oplus vartheta 1 H oplus n 1 where ϑ 1 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 Edit There is a space C P displaystyle mathbf CP infty which in a sense is the inductive limit of C P n displaystyle mathbf CP n as n displaystyle n to 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 mapsS 1 S 2 n 1 C P n displaystyle S 1 hookrightarrow S 2n 1 twoheadrightarrow mathbf CP n and n displaystyle n to infty This gives a fiber bundle called the universal circle bundle S 1 S C P displaystyle S 1 hookrightarrow S infty twoheadrightarrow mathbf CP infty constructing this space Note using the long exact sequence of homotopy groups we have p 2 C P p 1 S 1 displaystyle pi 2 mathbf CP infty pi 1 S 1 hence C P displaystyle mathbf CP infty is an Eilenberg MacLane space a K Z 2 displaystyle K mathbb Z 2 Because of this fact and Brown s representability theorem we have the following isomorphism H 2 X Z X C P displaystyle H 2 X mathbb Z cong X mathbf CP infty for any nice CW complex X displaystyle X Moreover from the theory of Chern classes every complex line bundle L X displaystyle L to X can be represented as a pullback of the universal line bundle on C P displaystyle mathbf CP infty meaning there is a pullback square L L X C P displaystyle begin matrix L amp to amp mathcal L downarrow amp amp downarrow X amp to amp mathbf CP infty end matrix where L C P displaystyle mathcal L to mathbf CP infty is the associated vector bundle of the principal U 1 displaystyle U 1 bundle S C P displaystyle S infty to mathbf CP infty See for instance Bott amp Tu 1982 and Milnor amp Stasheff 1974 Differential geometry EditThe 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 P 1 U n P U n displaystyle mathrm P 1 times mathrm U n cong mathrm PU n It is a Hermitian symmetric space Kobayashi amp Nomizu 1996 represented as a coset space U n 1 U 1 U n S U n 1 S U 1 U n 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 Edit 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 Edit 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 amp Schoen 2008 Spin structure Edit The odd dimensional projective spaces can be given a spin structure the even dimensional ones cannot Algebraic geometry EditComplex projective space is a special case of a Grassmannian and is a homogeneous space for various Lie groups It is a Kahler 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 amp Harris 1994 Zariski topology Edit Main article 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 Zn denote the commutative ring of polynomials in the n 1 variables Z0 Zn This ring is graded by the total degree of each polynomial S n 0 S n displaystyle S bigoplus n 0 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 Edit Another construction of CPn and its Zariski topology is possible Let S S be the ideal spanned by the homogeneous polynomials of positive degree n gt 0 S n displaystyle bigoplus n gt 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 V I p Proj S p I displaystyle V I p in operatorname 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 Edit 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 f l z l k f z displaystyle f lambda z lambda k f z for all l 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 lv V for all l 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 C n 1 C P n C P n displaystyle mathbf C n 1 times mathbf CP n to mathbf CP n whose fiber over L CPn is the set x L x L 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 kth 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 O k H O k 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 Poincare 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 See also EditGromov s inequality for complex projective space Projective Hilbert space Quaternionic projective space Real projective space Complex affine space K3 surfaceReferences EditBesse Arthur L 1978 Manifolds all of whose geodesics are closed Ergebnisse der Mathematik und ihrer Grenzgebiete Results in Mathematics and Related Areas vol 93 Berlin New York Springer Verlag ISBN 978 3 540 08158 6 Bott Raoul Tu Loring W 1982 Differential Forms in Algebraic Topology Berlin New York Springer Verlag ISBN 978 0 387 90613 3 Brendle Simon Schoen Richard 2008 Classification of manifolds with weakly 1 4 pinched curvatures Acta Mathematica 200 1 13 arXiv 0705 3963 doi 10 1007 s11511 008 0022 7 Grattan Guinness Ivor 2005 Landmark writings in western mathematics 1640 1940 Elsevier ISBN 978 0 444 50871 3 Griffiths Phillip Harris Joseph 1994 Principles of algebraic geometry Wiley Classics Library New York John Wiley amp Sons ISBN 978 0 471 05059 9 MR 1288523 Hartshorne Robin 1977 Algebraic Geometry Graduate Texts in Mathematics vol 52 New York Springer Verlag ISBN 978 0 387 90244 9 MR 0463157 Klingenberg Wilhelm 1982 Riemannian geometry Walter de Greuter ISBN 978 3 11 008673 7 Kobayashi Shoshichi Nomizu Katsumi 1996 Foundations of Differential Geometry Volume II Wiley Classics Library edition ISBN 978 0 471 15732 8 Milnor John Willard Stasheff James D 1974 Characteristic classes Princeton University Press MR 0440554 von Staudt Karl Georg Christian 1860 Beitrage zur Geometrie der Lage Nuremberg Retrieved from https en wikipedia org w index php title Complex projective space amp oldid 1132242911, wikipedia, wiki, book, books, library,

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