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| [[File:Stereographic projection in 3D.png|thumb|right|<center>The [[Riemann sphere]], the one-dimensional complex projective space, i.e. the [[complex projective line]].</center>]]
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| In [[mathematics]], '''complex projective space''' is the [[projective space]] with respect to the field of [[complex number]]s. 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 plane|complex]]'' lines through the origin of a complex Euclidean space (see [[#Introduction|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'''('''C'''<sup>''n''+1</sup>), '''P'''<sub>''n''</sub>('''C''') or '''CP'''<sup>''n''</sup>. When {{nowrap|''n'' {{=}} 1}}, the complex projective space '''CP'''<sup>1</sup> is the [[Riemann sphere]], and when {{nowrap|''n'' {{=}} 2}}, '''CP'''<sup>2</sup> is the [[complex projective plane]] (see there for a more elementary discussion).
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| Complex projective space was first introduced by {{harvtxt|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 variety|algebraic varieties]] {{harv|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 [[N-sphere|sphere]]. Indeed, in a certain sense the (2''n''+1)-sphere can be regarded as a family of circles parametrized by '''CP'''<sup>''n''</sup>: this is the [[Hopf fibration]]. Complex projective space carries a ([[Kähler metric|Kähler]]) [[metric tensor|metric]], called the [[Fubini–Study metric]], in terms of which it is a [[Hermitian symmetric space]] of rank 1.
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| Complex projective space has many applications in both mathematics and [[quantum physics]]. In [[algebraic geometry]], complex projective space is the home of [[projective variety|projective varieties]], a well-behaved class of [[algebraic variety|algebraic varieties]]. In topology, the complex projective space plays an important role as a [[classifying space]] for complex [[line bundle]]s: families of complex lines parametrized by another space. In this context, the infinite union of projective spaces ([[direct limit]]), denoted '''CP'''<sup>∞</sup>, is the classifying space [[K(Z,2)]]. In quantum physics, the wave function associated to a [[pure state]]s 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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| ==Introduction==
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| [[File:Railroad-Tracks-Perspective.jpg|thumb|right|Parallel lines in the plane intersect at the [[vanishing point]] in the line at infinity.]]
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| The notion of a projective plane arises out of the idea of perspective 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.
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| These [[real projective space]]s can be constructed in a slightly more rigorous way as follows. Here, let '''R'''<sup>''n''+1</sup> 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 '''R'''<sup>''n''+1</sup>. 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'''<sup>''n''+1</sup>. Without reference to coordinates, this is the space of lines through the origin in an (''n''+1)-dimensional real [[vector space]].
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| 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'''<sup>''n''+1</sup> (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'''<sup>''n''</sup>) 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'''<sup>''n''+1</sup>, where two directions are regarded as the same if they differ by a phase.
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| ==Construction==
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| Complex projective space is a [[complex manifold]] that may be described by ''n'' + 1 complex coordinates as
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| :<math>Z=(Z_1,Z_2,\ldots,Z_{n+1}) \in \mathbb{C}^{n+1},
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| \qquad (Z_1,Z_2,\ldots,Z_{n+1})\neq (0,0,\ldots,0)</math>
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| where the tuples differing by an overall rescaling are identified:
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| :<math>(Z_1,Z_2,\ldots,Z_{n+1}) \equiv
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| (\lambda Z_1,\lambda Z_2, \ldots,\lambda Z_{n+1});
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| \quad \lambda\in \mathbb{C},\qquad \lambda \neq 0.</math>
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| That is, these are [[homogeneous coordinates]] in the traditional sense of [[projective geometry]]. The point set '''CP'''<sup>''n''</sup> is covered by the patches <math>U_i=\{ Z \mid Z_i\ne0\}</math>. In ''U''<sub>''i''</sub>, one can define a coordinate system by
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| :<math>z_1 = Z_1/Z_i, z_2=Z_2/Z_i, \dots, z_{i-1}=Z_{i-1}/Z_i, z_i = Z_{i+1}/Z_i,\dots, z_n=Z_{n+1}/Z_i.</math> | |
| The coordinate transitions between two different such charts ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> are [[holomorphic function]]s (in fact they are [[fractional linear transformation]]s). Thus '''CP'''<sup>''n''</sup> carries the structure of a [[complex manifold]] of complex dimension ''n'', and ''[[a fortiori]]'' the structure of a real [[differentiable manifold]] of real dimension 2''n''.
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| One may also regard '''CP'''<sup>''n''</sup> as a [[quotient space|quotient]] of the unit 2''n'' + 1 [[sphere]] in '''C'''<sup>''n''+1</sup> under the action of [[Unitary group|U(1)]]:
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| :'''CP'''<sup>''n''</sup> = ''S''<sup>2''n''+1</sup>/U(1). | |
| This is because every line in '''C'''<sup>''n''+1</sup> 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 '''CP'''<sup>''n''</sup>. For ''n'' = 1 this construction yields the classical [[Hopf bundle]] <math>S^3\to S^2</math>. From this perspective, the differentiable structure on '''CP'''<sup>''n''</sup> is induced from that of ''S''<sup>2''n''+1</sup>, being the quotient of the latter by a compact group that acts properly.
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| ==Topology==
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| The topology of '''CP'''<sup>''n''</sup> is determined inductively by the following [[CW complex|cell decomposition]]. Let ''H'' be a fixed hyperplane through the origin in '''C'''<sup>''n''+1</sup>. Under the projection map {{nowrap|'''C'''<sup>''n''+1</sup>\{0} → '''CP'''<sup>''n''</sup>}}, ''H'' goes into a subspace that is homeomorphic to '''CP'''<sup>''n''−1</sup>. The complement of ''H'' in '''CP'''<sup>''n''</sup> is homeomorphic to '''C'''<sup>''n''</sup>. Thus '''CP'''<sup>''n''</sup> arises by attaching a 2''n''-cell to '''CP'''<sup>''n''−1</sup>:
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| :<math>\mathbf{CP}^n = \mathbf{CP}^{n-1}\cup \mathbf{C}^n.</math>
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| Alternatively, if the 2''n''-cell is regarded instead as the open unit ball in '''C'''<sup>''n''</sup>, 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 {{harv|Besse|1978}}.
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| ===Point-set topology===
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| Complex projective space is [[compact space|compact]] and [[connected space|connected]], being a quotient of a compact, connected space.
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| ===Homotopy groups===
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| From the fiber bundle
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| :<math>S^1 \hookrightarrow S^{2n+1} \twoheadrightarrow \mathbf{CP}^n</math>
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| or more suggestively
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| :<math>U(1) \hookrightarrow S^{2n+1} \twoheadrightarrow \mathbf{CP}^n</math>
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| '''CP'''<sup>''n''</sup> is [[simply connected]]. Moreover, by the [[long exact homotopy sequence]], the second homotopy group is {{nowrap|1=π<sub>2</sub>('''CP'''<sup>''n''</sup>) ≅ '''Z'''}}, and all the higher homotopy groups agree with those of ''S''<sup>2''n''+1</sup>: {{nowrap|1=π<sub>''k''</sub>('''CP'''<sup>''n''</sup>) ≅ π<sub>''k''</sub>(''S''<sup>2''n''+1</sup>)}} for all ''k'' > 2.
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| ===Homology===
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| In general, the [[algebraic topology]] of '''CP'''<sup>''n''</sup> is based on the rank of the [[homology group]]s being zero in odd dimensions; also ''H''<sub>2''i''</sub>('''CP'''<sup>''n''</sup>, '''Z''') is [[infinite cyclic]] for ''i'' = 0 to ''n''. Therefore the [[Betti number]]s run
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| :1, 0, 1, 0, ..., 0, 1, 0, 0, 0, ...
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| That is, 0 in odd dimensions, 1 in even dimensions up to 2n. The [[Euler characteristic]] of '''CP'''<sup>''n''</sup> is therefore ''n'' + 1. By [[Poincaré duality]] the same is true for the ranks of the [[cohomology group]]s. In the case of cohomology, one can go further, and identify the [[graded ring]] structure, for [[cup product]]; the generator of ''H''<sup>2</sup>('''CP'''<sup>n</sup>, '''Z''') is the class associated to a [[hyperplane]], and this is a ring generator, so that the ring is isomorphic with
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| :'''Z'''[''T'']/(''T''<sup>''n''+1</sup>),
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| with ''T'' a degree two generator. This implies also that the [[Hodge number]] ''h''<sup>''i'',''i''</sup> = 1, and all the others are zero. See {{harv|Besse|1978}}.
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| ===''K''-theory===
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| It follows from induction and [[Bott periodicity]] that
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| :<math>K_\mathbf{C}^*(\mathbf{CP}^n) = K_\mathbf{C}^0(\mathbf{CP}^n) = \mathbf{Z}[H]/(H-1)^{n+1}.</math>
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| The [[tangent bundle]] satisfies
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| :<math>T\mathbf{CP}^n \oplus \vartheta^1 = H^{\oplus n+1},</math>
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| where <math>\vartheta^1</math> denotes the trivial line bundle. From this, the [[Chern class]]es and [[characteristic number]]s can be calculated.
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| ===Classifying space===
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| There is a space '''CP'''<sup>∞</sup> which, in a sense, is the [[inductive limit]] of '''CP'''<sup>''n''</sup> as ''n'' → ∞. It is [[BU(1)]], the [[classifying space]] of [[U(1)]], in the sense of [[homotopy theory]], and so classifies complex [[line bundle]]s; equivalently it accounts for the first [[Chern class]]. See, for instance, {{harv|Bott|Tu|1982}} and {{harv|Milnor|Stasheff|1974}}. The space '''CP'''<sup>∞</sup> is also the same as the infinite-dimensional [[projective unitary group]]; see that article for additional properties and discussion.
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| ==Differential geometry==
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| The natural metric on '''CP'''<sup>''n''</sup> is the [[Fubini-Study metric]], and its isometry group is the [[projective unitary group]] PU(''n''+1), where the stabilizer of a point is
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| :<math>\mathrm{P}(1\times \mathrm{U}(n)) \cong \mathrm{PU}(n).</math>
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| It is a [[Hermitian symmetric space]] {{harv|Kobayashi|Nomizu|1996}}, represented as a coset space
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| :<math>U(n+1)/(U(1) \times U(n)) \cong SU(n+1)/S(U(1) \times U(n)).</math>
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| 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''.
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| ===Geodesics===
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| Through any two points ''p'', ''q'' in complex projective space, there passes a unique ''complex'' line (a '''CP'''<sup>1</sup>). 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.)
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| The [[cut locus]] of any point ''p'' is equal to a hyperplane '''CP'''<sup>''n''−1</sup>. This is also the set of fixed points of the geodesic symmetry at ''p'' (less ''p'' itself). See {{harv|Besse|1978}}.
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| ===Sectional curvature pinching===
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| 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 [[Riemannian_geometry#Pinched_sectional_curvature|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 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]] F<sub>4</sub>/Spin(9); see {{harv|Brendle-Schoen|2008}}.
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| ==Algebraic geometry==
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| Complex projective space is a special case of a [[Grassmannian]], and is a [[homogeneous space]] for various [[Lie group]]s. 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 [[Algebraic geometry and analytic geometry#Chow.27s theorem|Chow's theorem]], any compact complex submanifold of '''CP'''<sup>''n''</sup> is the zero locus of a finite number of polynomials, and is thus a projective [[algebraic variety]]. See {{harv|Griffiths|Harris|1994}}
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| ===Zariski topology===
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| In [[algebraic geometry]], complex projective space can be equipped with another topology known as the [[Zariski topology]] {{harv|Hartshorne|1971|loc=§II.2}}. Let {{nowrap|''S'' {{=}} '''C'''[''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>]}} denote the [[commutative ring]] of polynomials in the (''n''+1) variables ''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>. This ring is [[graded ring|graded]] by the total degree of each polynomial:
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| :<math>S = \bigoplus_{n=0}^\infty S_n.</math>
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| Define a subset of '''CP'''<sup>''n''</sup> 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 '''CP'''<sup>''n''</sup>.
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| ===Structure as a scheme===
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| Another construction of '''CP'''<sup>''n''</sup> (and its Zariski topology) is possible. Let ''S''<sub>+</sub> ⊂ ''S'' be the [[ideal (ring theory)|ideal]] spanned by the homogeneous polynomials of positive degree:
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| :<math>\bigoplus_{n>0}S_n.</math>
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| Define [[Proj|Proj ''S'']] to be the set of all [[homogeneous ideal|homogeneous]] [[prime ideal]]s in ''S'' that do not contain ''S''<sub>+</sub>. Call a closed of Proj ''S'' open if it has the form
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| :<math>V(I) = \{ p\in \operatorname{Proj} S\mid p\supseteq I\}</math>
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| for some ideal ''I'' in ''S''. The complements of these closed sets define a topology on Proj ''S''. The ring ''S'', by [[localization of a ring|localization at a prime ideal]], determines a [[sheaf (mathematics)|sheaf]] of [[local ring]]s on Proj ''S''. The space Proj ''S'', together with its topology and sheaf of local rings, is a [[scheme (mathematics)|scheme]]. The subset of closed points of Proj ''S'' is homeomorphic to '''CP'''<sup>''n''</sup> with its Zariski topology. Local sections of the sheaf are identified with the [[rational function]]s of total degree zero on '''CP'''<sup>''n''</sup>.
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| ===Line bundles===
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| All line bundles on complex projective space can be obtained by the following construction. A function {{nowrap|''f'' : '''C'''<sup>''n''+1</sup>\{0} → '''C'''}} is called [[homogeneous function|homogeneous]] of degree ''k'' if
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| :<math>f(\lambda z) = \lambda^k f(z)</math>
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| for all {{nowrap|λ ∈ '''C'''\{0}}} and {{nowrap|''z'' ∈ '''C'''<sup>''n''+1</sup>\{0}}}. More generally, this definition makes sense in [[cone (linear algebra)|cones]] in {{nowrap|'''C'''<sup>''n''+1</sup>\{0}}}. A set {{nowrap|''V'' ⊂ '''C'''<sup>''n''+1</sup>\{0}}} is called a cone if, whenever {{nowrap|''v'' ∈ ''V''}}, then {{nowrap|''λv'' ∈ ''V''}} for all {{nowrap|λ ∈ '''C'''\{0}}}; that is, a subset is a cone if it contains the complex line through each of its points. If {{nowrap|''U'' ⊂ '''CP'''<sup>''n''</sup>}} is an open set (in either the analytic topology or the [[Zariski topology]]), let {{nowrap|''V'' ⊂ '''C'''<sup>''n''+1</sup>\{0}}} be the cone over ''U'': the preimage of ''U'' under the projection {{nowrap|'''C'''<sup>''n''+1</sup>\{0} → '''CP'''<sup>''n''</sup>}}. 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 (mathematics)|sheaf]] of sections of a certain line bundle, denoted by ''O''(''k'').
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| In the special case {{nowrap|''k'' {{=}} −1}}, the bundle ''O''(−1) is called the [[tautological line bundle]]. It is equivalently defined as the subbundle of the product
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| :<math>\mathbf{C}^{n+1}\times\mathbf{CP}^n\to \mathbf{CP}^n</math>
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| whose fiber over {{nowrap|''L'' ∈ '''CP'''<sup>''n''</sup>}} is the set
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| :<math>\{(x,L)\mid x\in L\}.</math>
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| These line bundles can also be described in the language of [[divisor (algebraic geometry)|divisors]]. Let ''H'' = '''CP'''<sup>''n''−1</sup> be a given complex hyperplane in '''CP'''<sup>''n''</sup>. The space of [[meromorphic function]]s on '''CP'''<sup>''n''</sup> 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 ''k''<sup>th</sup> 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
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| :<math>O(kH) \cong O(k).</math>
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| Indeed, if {{nowrap|''L''(''z'') {{=}} 0}} is a linear defining function for ''H'', then ''L''<sup>−''k''</sup> is a meromorphic section of ''O''(''k''), and locally the other sections of ''O''(''k'') are multiples of this section.
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| Since {{nowrap|''H''<sup>1</sup>('''CP'''<sup>''n''</sup>,'''Z''') {{=}} 0}}, the line bundles on '''CP'''<sup>''n''</sup> are classified up to isomorphism by their [[Chern class]]es, which are integers: they lie in {{nowrap|''H''<sup>2</sup>('''CP'''<sup>''n''</sup>,'''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 '''CP'''<sup>''n''</sup> is a tensor power of ''O''(''H'') or ''O''(−''H''). In other words, the [[Picard group]] of '''CP'''<sup>''n''</sup> is generated as an abelian group by the hyperplane class [''H''] {{harv|Hartshorne|1977}}.
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| ==See also==
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| * [[Gromov's inequality for complex projective space]]
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| * [[Projective Hilbert space]]
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| * [[Quaternionic projective space]]
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| * [[Real projective space]]
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| ==References==
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| * {{Citation | last1=Besse | first1=Arthur L. | author1-link=Arthur Besse |title=Manifolds all of whose geodesics are closed | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas] | isbn=978-3-540-08158-6 | year=1978 | volume=93}}.
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| * {{Citation | last1=Bott | first1=Raoul | author1-link=Raoul Bott | last2=Tu | first2=Loring W. | title=Differential Forms in Algebraic Topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90613-3 | year=1982}}.
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| *{{Citation | last1=Brendle | first1=Simon | last2=Schoen | first2=Richard | title=Classification of manifolds with weakly 1/4-pinched curvatures | year=2008 | journal=[[Acta Mathematica]] | volume=200 | pages=1–13|url=
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| http://link.springer.com/content/pdf/10.1007%2Fs11511-008-0022-7}}. | |
| * {{citation |last=Grattan-Guinness |first=Ivor |title=Landmark writings in western mathematics 1640–1940 |year=2005 |publisher=Elsevier |isbn=0-444-50871-6}}.
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| * {{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | mr=1288523 | year=1994}}.
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| * {{Hartshorne AG}}
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| * {{citation |first=Wilhelm |last=Klingenberg |authorlink=Wilhelm Klingenberg |title=Riemannian geometry |publisher=Walter de Greuter |year=1982 |isbn=3-11-008673-5}}.
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| * {{citation |first=Shoshichi |last= Kobayashi |first2= Katsumi |last2= Nomizu |title=[[Foundations of Differential Geometry]], Volume II |publisher= Wiley Classics Library edition |year= 1996 |isbn= 0-471-15732-5}}.
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| * {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | last2=Stasheff | first2=James D. |author2-link=Jim Stasheff| title=Characteristic classes | publisher=[[Princeton University Press]] | mr=0440554 | year=1974}}.
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| * {{citation |first=Karl Georg Christian |last=von Staudt |authorlink=Karl Georg Christian von Staudt |year=1860 |title=Beiträge zur Geometrie der Lage |publication-place=Nuremberg}}.
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| {{DEFAULTSORT:Complex Projective Space}}
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| [[Category:Algebraic varieties]]
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| [[Category:Complex manifolds]]
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| [[Category:Projective geometry]]
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