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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 mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
in the plane having a
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
line at each point determines a varying line: the ''
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
'' is a way of organising these. More formally, in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
and differential topology, a line bundle is defined as a ''
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1
invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
real matrices, which is
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
-equivalent to a
discrete two-point space In topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle. From the perspective of homotopy theory, a real line bundle therefore behaves much the same as a fiber bundle with a two-point fiber, that is, like a double cover. A special case of this is the
orientable double cover In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, Surface (topology), surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclo ...
of a differentiable manifold, where the corresponding line bundle is the determinant bundle of the tangent bundle (see below). The Möbius strip corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
as a fiber, or the real line. Complex line bundles are closely related to
circle bundle In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1. Oriented circle bundles are also known as principal ''U''(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circ ...
s. There are some celebrated ones, for example the
Hopf fibration In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz H ...
s of
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
s to spheres. In algebraic geometry, an
invertible sheaf In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
(i.e.,
locally free sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
of rank one) is often called a line bundle. Every line bundle arises from a divisor with the following conditions (I) If ''X'' is reduced and irreducible scheme, then every line bundle comes from a divisor. (II) If ''X'' is projective scheme then the same statement holds.


The tautological bundle on projective space

One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space. The projectivization P(''V'') of a vector space ''V'' over a field ''k'' is defined to be the quotient of V \setminus \ by the action of the multiplicative group ''k''×. Each point of P(''V'') therefore corresponds to a copy of ''k''×, and these copies of ''k''× can be assembled into a ''k''×-bundle over P(''V''). ''k''× differs from ''k'' only by a single point, and by adjoining that point to each fiber, we get a line bundle on P(''V''). This line bundle is called the tautological line bundle. This line bundle is sometimes denoted \mathcal(-1) since it corresponds to the dual of the Serre twisting sheaf \mathcal(1).


Maps to projective space

Suppose that ''X'' is a space and that ''L'' is a line bundle on ''X''. A global section of ''L'' is a function such that if is the natural projection, then = id''X''. In a small neighborhood ''U'' in ''X'' in which ''L'' is trivial, the total space of the line bundle is the product of ''U'' and the underlying field ''k'', and the section ''s'' restricts to a function . However, the values of ''s'' depend on the choice of trivialization, and so they are determined only up to multiplication by a nowhere-vanishing function. Global sections determine maps to projective spaces in the following way: Choosing not all zero points in a fiber of ''L'' chooses a fiber of the tautological line bundle on P''r'', so choosing non-simultaneously vanishing global sections of ''L'' determines a map from ''X'' into projective space P''r''. This map sends the fibers of ''L'' to the fibers of the dual of the tautological bundle. More specifically, suppose that are global sections of ''L''. In a small neighborhood ''U'' in ''X'', these sections determine ''k''-valued functions on ''U'' whose values depend on the choice of trivialization. However, they are determined up to ''simultaneous'' multiplication by a non-zero function, so their ratios are well-defined. That is, over a point ''x'', the values are not well-defined because a change in trivialization will multiply them each by a non-zero constant λ. But it will multiply them by the ''same'' constant λ, so the
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
's''0(''x'') : ... : ''s''''r''(''x'')are well-defined as long as the sections do not simultaneously vanish at ''x''. Therefore, if the sections never simultaneously vanish, they determine a form 's''0 : ... : ''s''''r''which gives a map from ''X'' to P''r'', and the pullback of the dual of the tautological bundle under this map is ''L''. In this way, projective space acquires a
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
. The universal way to determine a map to projective space is to map to the projectivization of the vector space of all sections of ''L''. In the topological case, there is a non-vanishing section at every point which can be constructed using a bump function which vanishes outside a small neighborhood of the point. Because of this, the resulting map is defined everywhere. However, the codomain is usually far, far too big to be useful. The opposite is true in the algebraic and holomorphic settings. Here the space of global sections is often finite dimensional, but there may not be any non-vanishing global sections at a given point. (As in the case when this procedure constructs a
Lefschetz pencil In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety ''V''. Description A ''pencil'' is a particular kind of linear system of ...
.) In fact, it is possible for a bundle to have no non-zero global sections at all; this is the case for the tautological line bundle. When the line bundle is sufficiently ample this construction verifies the
Kodaira embedding theorem In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomial ...
.


Determinant bundles

In general if ''V'' is a vector bundle on a space ''X'', with constant fibre dimension ''n'', the ''n''-th
exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
of ''V'' taken fibre-by-fibre is a line bundle, called the determinant line bundle. This construction is in particular applied to the cotangent bundle of a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. The resulting determinant bundle is responsible for the phenomenon of
tensor densities In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it ...
, in the sense that for an
orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
it has a nonvanishing global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
. The same construction (taking the top exterior power) applies to a finitely generated projective module ''M'' over a Noetherian domain and the resulting invertible module is called the determinant module of ''M''.


Characteristic classes, universal bundles and classifying spaces

The first
Stiefel–Whitney class In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of ...
classifies smooth real line bundles; in particular, the collection of (equivalence classes of) real line bundles are in correspondence with elements of the first cohomology with Z/2Z coefficients; this correspondence is in fact an isomorphism of abelian groups (the group operations being tensor product of line bundles and the usual addition on cohomology). Analogously, the first
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
classifies smooth complex line bundles on a space, and the group of line bundles is isomorphic to the second cohomology class with integer coefficients. However, bundles can have equivalent
smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...
s (and thus the same first Chern class) but different holomorphic structures. The Chern class statements are easily proven using the
exponential sequence In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let ''M'' be a complex manifold, and write ''O'M'' for the sheaf of holomorphic functions on ''M''. Let ''O'M''* be ...
of sheaves on the manifold. One can more generally view the classification problem from a homotopy-theoretic point of view. There is a universal bundle for real line bundles, and a universal bundle for complex line bundles. According to general theory about
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
s, the heuristic is to look for
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
spaces on which there are
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
s of the respective groups ''C''2 and ''S''1, that are free actions. Those spaces can serve as the universal
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s, and the quotients for the actions as the classifying spaces ''BG''. In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex projective space. Therefore the classifying space ''BC''2 is of the homotopy type of RP, the real projective space given by an infinite sequence of
homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...
. It carries the universal real line bundle; in terms of homotopy theory that means that any real line bundle ''L'' on a
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
''X'' determines a ''classifying map'' from ''X'' to RP, making ''L'' a bundle isomorphic to the pullback of the universal bundle. This classifying map can be used to define the Stiefel-Whitney class of ''L'', in the first cohomology of ''X'' with Z/2Z coefficients, from a standard class on RP. In an analogous way, the complex projective space CP carries a universal complex line bundle. In this case classifying maps give rise to the first
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
of ''X'', in H2(''X'') (integral cohomology). There is a further, analogous theory with quaternionic (real dimension four) line bundles. This gives rise to one of the
Pontryagin class In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundl ...
es, in real four-dimensional cohomology. In this way foundational cases for the theory of
characteristic class In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes ...
es depend only on line bundles. According to a general
splitting principle In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computati ...
this can determine the rest of the theory (if not explicitly). There are theories of
holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
s on complex manifolds, and
invertible sheaves In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
in algebraic geometry, that work out a line bundle theory in those areas.


See also

* I-bundle *
Ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...


Notes


References

* Michael Murray
Line Bundles
2002 (PDF web link) *
Robin Hartshorne __NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
.
Algebraic geometry
'. AMS Bookstore, 1975. {{DEFAULTSORT:Line Bundle Differential topology Algebraic topology Homotopy theory Vector bundles