Complex projective space

Complex projective space

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, complex projective space is the projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

with respect to the field of complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s. By analogy, whereas the points of a real projective space
Real projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...

label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

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
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

. The space is denoted variously as P(Cn+1), Pn(C) or CPn. When , the complex projective space CP1 is the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

, and when , CP2 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 Carnot
Lazare Nicolas Marguerite, Comte Carnot , the Organizer of Victory in the French Revolutionary Wars, was a French politician, engineer, and mathematician.-Education and early life:...

, a kind of synthetic geometry
Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...

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
Italian school of algebraic geometry
In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...

that the complex projective spaces were the most natural domains in which to consider the solutions of polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

equations—algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

. In modern times, both the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

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
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

, called the Fubini–Study metric, in terms of which it is a Hermitian symmetric space
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...

of rank 1.

Complex projective space has many applications in both mathematics and quantum physics. In algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, complex projective space is the home of projective varieties, a well-behaved class of algebraic varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

. In topology, the complex projective space plays an important role as a classifying space
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 by a free action of G...

for complex line bundle
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 organising these...

s: families of complex lines parametrized by another space. In this context, the infinite union of projective spaces (direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...

), denoted CP, is the classifying space K(Z,2). In quantum physics, the wave function associated to a pure states of a quantum mechanical system is a probability amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...

, 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
Projective Hilbert space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space P of a complex Hilbert space H is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by...

of the state space.

Introduction

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 projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

, 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 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 4-dimensions) would see.

These are, properly speaking, the real projective space
Real projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...

s. They 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
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

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
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

.

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
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

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
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

in the traditional sense of projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

. 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 function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

s (in fact they are fractional linear transformations). Thus CPn carries the structure of a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

of complex dimension n, and a fortiori the structure of a real differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

of real dimension 2n.

One may also regard CPn as a quotient
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

of the unit 2n + 1 sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

in Cn+1 under the action of U(1)
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...

:
CPn = S2n+1/U(1).

This is because every line in Cn+1 intersects the unit sphere in a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

. 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
Hopf bundle
In the mathematical field of topology, the Hopf fibration describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber 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
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...

. Let H be a fixed hyperplane through the origin in Cn+1. Under the projection map , H goes into a subspace that is homeomorphic to CPn−1. The complement 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 .

Point-set topology

Complex projective space is compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

and connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

, 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 , and all the higher homotopy groups agree with those of S2n+1: for all k > 2.

Homology

In general, the algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which 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 up to homotopy equivalence.Although 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 number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

s 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
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

of CPn is therefore n + 1. By Poincaré duality
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...

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
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 graded commutative product operation in cohomology, turning the cohomology of a space X into a...

; the generator of H2(CPn, Z) is the class associated to a hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

, 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 .

K-theory

It follows from induction and Bott periodicity that

The tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

satisfies

where denotes the trivial line bundle. From this, the Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

es and characteristic numbers can be calculated.

Classifying space

There is a space CP which, in a sense, is the limit of CPn as n → ∞. It is BU(1), the classifying space
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 by a free action of G...

of U(1), in the sense of homotopy theory, and so classifies complex line bundle
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 organising these...

s; equivalently it accounts for the first Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

. See, for instance, and . The space CP is also the same as the infinite-dimensional projective unitary group
Projective unitary group
In mathematics, the projective unitary group PU is the quotient of the unitary group U by the right multiplication of its center, U, embedded as scalars....

; see that article for additional properties and discussion.

Geometry

The natural metric on CPn is the Fubini-Study metric
Fubini-Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study....

, and its isometry group is the projective unitary group
Projective unitary group
In mathematics, the projective unitary group PU is the quotient of the unitary group U by the right multiplication of its center, U, embedded as scalars....

PU(n+1), where the stabilizer of a point is

It is a Hermitian symmetric space
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...

, 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
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

of this complex line that contains p and q is a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...

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 .

Sectional curvature pinching

It has sectional curvature
Sectional curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

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 homeomorphic to the sphere. Complex projective space shows that 1/4 is sharp. Conversely, any complete simply connected δ-pinched Riemannian manifold is either homeomorphic to the sphere, or diffeomorphic to the complex projective space, the quaternionic projective space
Quaternionic projective space
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted byand is a closed manifold of dimension 4n...

, or else the Cayley plane
Cayley plane
In mathematics, the Cayley plane OP2 is a projective plane over the octonions. It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley ....

F4/Spin(9); see .

Algebraic geometry

Complex projective space is a special case of a Grassmannian
Grassmannian
In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...

, and is a homogeneous space
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...

for various Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s. It is a Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

carrying the Fubini-Study metric
Fubini-Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study....

, which is essentially determined by symmetry properties. It also plays a central role in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

; 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
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

. See

Zariski topology

In algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, complex projective space can be equipped with another topology known as the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...

. Let denote the commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

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
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

spanned by the homogeneous polynomials of positive degree:
Define Proj S to be the set of all homogeneous prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

s in S that do not contain S+. Call a closed of Proj S open 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
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

, determines a sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

of local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...

s on Proj S. The space Proj S, together with its topology and sheaf of local rings, is a scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

. 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 function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

s of total degree zero on CPn.

Line bundles

All line bundles on complex projective space can be obtained by the following construction. A function is called homogeneous
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

of degree k if
for all } and }. More generally, this definition makes sense in cones
Cone (linear algebra)
In linear algebra, a cone is a subset of a vector space that is closed under multiplication by positive scalars. In other words, a subset C of a real vector space V is a cone if and only if λx belongs to C for any x in C and any positive scalar λ of V .A cone is said...

in }. A set } is called a cone if, whenever , then for all }; that is, a subset is a cone if it contains the complex line through each of its points. If is an open set (in either the analytic topology or the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...

), let } be the cone over U: the preimage of U under the projection . 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
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

of sections of a certain line bundle, denoted by O(k).

In the special case , the bundle O(−1) is called the tautological line bundle. It is equivalently defined as the subbundle of the product
whose fiber over is the set

These line bundles can also be described in the language of divisors
Divisor (algebraic geometry)
In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors...

. Let H = CPn−1 be a given complex hyperplane in CPn. The space of meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

s on CPn−1 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 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 , the line bundles on CPn are classified up to isomorphism by their Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...

es, which are integers: they lie in . In fact, the first Chern classes of complex projective space are generated under Poincaré duality
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...

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] .