Tautological bundle

In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of formula_1-dimensional subspaces of formula_2, given a point in the Grassmannian corresponding to a formula_1-dimensional vector subspace formula_4, the fiber over formula_5 is the subspace formula_5 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 formula_8. The hyperplane bundle is the line bundle corresponding to the hyperplane (divisor) formula_9 in formula_10. 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.
Intuitive definition.
Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space formula_5. If formula_12 is a Grassmannian, and formula_13 is the subspace of formula_5 corresponding to formula_15 in formula_12, this is already almost the data required for a vector bundle: namely a vector space for each point formula_15, varying continuously. All that can stop the definition of the tautological bundle from this indication, is the difficulty that the formula_13 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 formula_13, that now do not intersect. With this, we have the bundle.
The projective space case is included. By convention formula_20 may usefully carry the tautological bundle in the dual space sense. That is, with formula_21 the dual space, points of formula_20 carry the vector subspaces of formula_21 that are their kernels, when considered as (rays of) linear functionals on formula_21. If formula_2 has dimension formula_26, the tautological line bundle is one tautological bundle, and the other, just described, is of rank formula_27.
Formal definition.
Let formula_28 be the Grassmannian of "n"-dimensional vector subspaces in formula_29 as a set it is the set of all "n"-dimensional vector subspaces of formula_30 For example, if "n" = 1, it is the real projective "k"-space.
We define the tautological bundle γ"n", "k" over formula_28 as follows. The total space of the bundle is the set of all pairs ("V", "v") consisting of a point "V" of the Grassmannian and a vector "v" in "V"; it is given the subspace topology of the Cartesian product formula_32 The projection map π is given by π("V", "v") = "V". If "F" is the pre-image of "V" under π, it is given a structure of a vector space by "a"("V", "v") + "b"("V", "w") = ("V", "av" + "bw"). Finally, to see local triviality, given a point "X" in the Grassmannian, let "U" be the set of all "V" such that the orthogonal projection "p" onto "X" maps "V" isomorphically onto "X", and then define
which is clearly a homeomorphism. Hence, the result is a vector bundle of rank "n".
The above definition continues to make sense if we replace formula_34 with the complex field formula_35
By definition, the infinite Grassmannian formula_36 is the direct limit of formula_28 as formula_38 Taking the direct limit of the bundles γ"n", "k" gives the tautological bundle γ"n" of formula_39 It is a universal bundle in the sense: for each compact space "X", there is a natural bijection
where on the left the bracket means homotopy class and on the right is the set of isomorphism classes of real vector bundles of rank "n". The inverse map is given as follows: since "X" is compact, any vector bundle "E" is a subbundle of a trivial bundle: formula_41 for some "k" and so "E" determines a map 
unique up to homotopy.
Remark: In turn, one can define a tautological bundle as a universal bundle; suppose there is a natural bijection
for any paracompact space "X". Since formula_36 is the direct limit of compact spaces, it is paracompact and so there is a unique vector bundle over formula_36 that corresponds to the identity map on formula_39 It is precisely the tautological bundle and, by restriction, one gets the tautological bundles over all formula_47
Hyperplane bundle.
The hyperplane bundle "H" on a real projective "k"-space is defined as follows. The total space of "H" is the set of all pairs ("L", "f") consisting of a line "L" through the origin in formula_48 and "f" a linear functional on "L". The projection map π is given by π("L", "f") = "L" (so that the fiber over "L" is the dual vector space of "L".) The rest is exactly like the tautological line bundle.
In other words, "H" is the dual bundle of the tautological line bundle.
In algebraic geometry, the hyperplane bundle is the line bundle (as invertible sheaf) corresponding to the hyperplane divisor
given as, say, "x"0 = 0, when "xi" are the homogeneous coordinates. This can be seen as follows. If "D" is a (Weil) divisor on formula_50 one defines the corresponding line bundle "O"("D") on "X" by 
where "K" is the field of rational functions on "X". Taking "D" to be "H", we have:
where "x"0 is, as usual, viewed as a global section of the twisting sheaf "O"(1). (In fact, the above isomorphism is part of the usual correspondence between Weil divisors and Cartier divisors.) Finally, the dual of the twisting sheaf corresponds to the tautological line bundle (see below).
Tautological line bundle in algebraic geometry.
In algebraic geometry, this notion exists over any field "k". The concrete definition is as follows. Let formula_53 and formula_54. Note that we have:
where Spec is relative Spec. Now, put:
where "I" is the ideal sheaf generated by global sections formula_57. Then "L" is a closed subscheme of formula_58 over the same base scheme formula_10; moreover, the closed points of "L" are exactly those ("x", "y") of formula_60 such that either "x" is zero or the image of "x" in formula_10 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 formula_62, where the locus "x" = 0 in "L" is the exceptional divisor. (cf. Hartshorne, Ch. I, the end of § 4.)
In general, formula_63 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 formula_65 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 formula_66 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.
Facts.
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.