Грассмандық кеңістіктегі тавтологиялық шоғырлама

In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of dimensional subspaces of , given a point in the Grassmannian corresponding to a dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is

the dual of the hyperplane bundle or Serre's twisting sheaf The hyperplane bundle is the line bundle corresponding to the hyperplane (divisor) in The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space. In Michael Atiyah's "K theory", the tautological line bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called the Hopf bundle. (cf. Bott generator.) More generally, there are also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle. The older term canonical bundle has dropped out of favour, on the grounds that canonical is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the canonical class in algebraic geometry could scarcely be avoided.

Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space If is a Grassmannian, and is the subspace of corresponding to in , this is already almost the data required for a vector bundle: namely a vector space for each point , varying continuously. All that can stop the definition of the tautological bundle from this indication, is the difficulty that the are going to intersect. Fixing this up is a routine application of the disjoint union device, so that the bundle projection is from a total space made up of identical copies of the , that now do not intersect. With this, we have the bundle. The projective space case is included. By convention may usefully carry the tautological bundle in the dual space sense. That is, with the dual space, points of carry the vector subspaces of that are their kernels, when considered as (rays of) linear functionals on If has dimension , the tautological line bundle is one tautological bundle, and the other, just described, is of rank .

In algebraic geometry, this notion exists over any field k. The concrete definition is as follows. Let and Note that we have:

where Spec is relative Spec. Now, put:

where I is the ideal sheaf generated by global sections Then L is a closed subscheme of over the same base scheme ; moreover, the closed points of L are exactly those (x, y) of such that either x is zero or the image of x in is y. Thus, L is the tautological line bundle as defined before if k is the field of real or complex numbers. In more concise terms, L is the blow up of the origin of the affine space , where the locus x = 0 in L is the exceptional divisor. (cf. Hartshorne, Ch. I, the end of § 4.) In general, is the algebraic vector bundle corresponding to a locally free sheaf E of finite rank. Since we have the exact sequence:

the tautological line bundle L, as defined above, corresponds to the dual of Serre's twisting sheaf. In practice both the notions (tautological line bundle and the dual of the twisting sheaf) are used interchangeably. Over a field, its dual line bundle is the line bundle associated to the hyperplane divisor H, whose global sections are the linear forms. Its Chern class is −H. This is an example of an anti ample line bundle. Over this is equivalent to saying that it is a negative line bundle, meaning that minus its Chern class is the de Rham class of the standard Kähler form.

The tautological line bundle γ1, k is locally trivial but not trivial, for k ≥ 1. This remains true over other fields. In fact, it is straightforward to show that, for k = 1, the real tautological line bundle is none other than the well known bundle whose total space is the Möbius strip. For a full proof of the above fact, see. The Picard group of line bundles on is infinite cyclic, and the tautological line bundle is a generator. In the case of projective space, where the tautological bundle is a line bundle, the associated invertible sheaf of sections is , the tensor inverse (ie the dual vector bundle) of the hyperplane bundle or Serre twist sheaf ; in other words the hyperplane bundle is the generator of the Picard group having positive degree (as a divisor) and the tautological bundle is its opposite: the generator of negative degree.