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In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: the fiber of the bundle over a vector space ''V'' (a point in the Grassmannian) is ''V'' itself. The dual of the tautological bundle has fiber over a vector space ''V'' that is the dual vector space of ''V''. 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〔Over a noncompact but paracompact base, this remains true provided one uses infinite Grassmannian.〕) 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) P''n''-1 in P''n''. The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space.〔In literature and textbooks, they are both often called canonical generators.〕 In 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.) 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. == Intuitive definition == Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space ''W''. If ''G'' is a Grassmannian, and ''V''''g'' is the subspace of ''W'' corresponding to ''g'' in ''G'', this is already almost the data required for a vector bundle: namely a vector space for each point ''g'', varying continuously. All that can stop the definition of the tautological bundle from this indication, is the (pedantic) difficulty that the ''V''''g'' 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 ''V''''g'', that now do not intersect. With this, we have the bundle. The projective space case is included. By convention and use ''P''(''V'') may usefully carry the tautological bundle in the dual space sense. That is, with ''V'' * the dual space, points of ''P''(''V'') carry the vector subspaces of ''V'' * that are their kernels, when considered as (rays of) linear functionals on ''V'' *. If ''V'' has dimension ''n'' + 1, the tautological line bundle is one tautological bundle, and the other, just described, is of rank ''n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tautological bundle」の詳細全文を読む スポンサード リンク
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