Because of this, the tautological bundle is important in the study of characteristic classes.
In algebraic geometry, the tautological line bundle (as invertible sheaf) is the dual of the hyperplane bundle or Serre's twisting sheaf
[2] In Michael Atiyah's "K-theory", the tautological line bundle over a complex projective space is called the standard line 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.
All that can stop the definition of the tautological bundle from this indication, is the difficulty that the
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
may usefully carry the tautological bundle in the dual space sense.
that are their kernels, when considered as (rays of) linear functionals on
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
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,[3] 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
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:
is the direct limit of compact spaces, it is paracompact and so there is a unique vector bundle over
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
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 algebraic geometry, the hyperplane bundle is the line bundle (as invertible sheaf) corresponding to the hyperplane divisor given as, say, x0 = 0, when xi are the homogeneous coordinates.
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 x0 is, as usual, viewed as a global section of the twisting sheaf O(1).
Finally, the dual of the twisting sheaf corresponds to the tautological line bundle (see below).
In algebraic geometry, this notion exists over any field k. The concrete definition is as follows.
Now, put: where I is the ideal sheaf generated by global sections
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
is the algebraic vector bundle corresponding to a locally free sheaf E of finite rank.
[4] Since we have the exact sequence: the tautological line bundle L, as defined above, corresponds to the dual
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.
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.
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.