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Wikipedia / 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. More formally, in algebraic topology and differential topology, a line bundle is defined as a vector bundle 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 real matrices, which is homotopy-equivalent to a discrete two-point space 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 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 as a fiber, or the real line.

Complex line bundles are closely related to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres.

In algebraic geometry, an invertible sheaf (i.e., locally free sheaf of rank one) is often called a line bundle.

Every line bundle arises from a divisor under the following conditions:

  1. is a reduced and irreducible scheme, then every line bundle comes from a divisor.

  2. is a 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. This line bundle is called the tautological line bundle since it corresponds to the dual of the Serre twisting sheaf.

Maps to projective space

. A global section depend on the choice of trivialization, and so they are determined only up to multiplication by a nowhere-vanishing function.

whose values depend on the choice of trivialization. However, they are determined up to simultaneous 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 this way, projective space acquires a universal property.

. 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 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.

Determinant bundles

-th exterior power 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. The resulting determinant bundle (more precisely, the bundle of a fixed nonegative power of the absolute values of its sections) is responsible for the phenomenon of tensor densities, in the sense that for an orientable manifold 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.

The same construction (taking the top exterior power) applies to a finitely generated projective module over a Noetherian domain and the resulting invertible module is called the determinant module.

Characteristic classes, universal bundles and classifying spaces

The first Stiefel–Whitney class 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 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 structures (and thus the same first Chern class) but different holomorphic structures. The Chern class statements are easily proven using the exponential sequence 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 spaces, the heuristic is to look for contractible spaces on which there are group actions, that are free actions. Those spaces can serve as the universal principal bundles. In these cases we can find those explicitly, in the infinite-dimensional analogues of real and complex projective space.

, the real projective space given by an infinite sequence of homogeneous coordinates on a CW complex determines a classifying map a bundle isomorphic to the pullback of the universal bundle. This classifying map can be used to define the Stiefel-Whitney class.

carries a universal complex line bundle. In this case classifying maps give rise to the first Chern class (integral cohomology).

There is a further, analogous theory with quaternionic (real dimension four) line bundles. This gives rise to one of the Pontryagin classes, in real four-dimensional cohomology.

In this way foundational cases for the theory of characteristic classes depend only on line bundles. According to a general splitting principle this can determine the rest of the theory (if not explicitly).

There are theories of holomorphic line bundles on complex manifolds, and invertible sheaves in algebraic geometry, that work out a line bundle theory in those areas.

See also

Notes

References