Tautological bundle

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.

Formal definition

Let G_n(R^n+k) be the Grassmannian of n-dimensional vector subspaces in R^n+k; as a set it is the set of all n-dimensional vector subspaces of R^n+k. For example, if n = 1, it is the real projective k-space.

We define the tautological bundle γ_{n, k} over G_n(R^n+k) 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 G_n(R^n+k) × R^n+k. 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

by ϕ (V, v) = (V, p(v)), 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 the field R by the complex field C.

By definition, the infinite Grassmannian G_n is the direct limit of G_n(R^n+k) as k →∞. Taking the direct limit of the bundles γ_{n, k} gives the tautological bundle γ_n of G_n. 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: E ↪ X × R n + k for some k and so E determines a map f E : X → G n , x ↦ E x , 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 G_n is the direct limit of compact spaces, it is paracompact and so there is a unique vector bundle over G_n that corresponds to the identity map on G_n. It is precisely the tautological bundle and, by restriction, one gets the tautological bundles over all G_n(R^n+k).

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 R^k+1 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, x₀ = 0, when x_i's are the homogeneous coordinates. This can be seen as follows. If D is a (Weil) divisor on X = Pⁿ, 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). (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 A = k [ y 0 , … , y n ] and P n = Proj ⁡ A . Note that we have:

where Spec is relative Spec. Now, put:

where I is the ideal sheaf generated by global sections x i y j − x j y i . Then L is a closed subscheme of A P n n + 1 over the same base scheme P n ; moreover, the closed points of L are exactly those (x, y) of A n + 1 × k P n such that either x is zero or the image of x in P n 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 A n + 1 , where the locus x = 0 in L is the exceptional divisor. (cf. Hartshorne, Ch. I, the end of § 4.)

In general, S p e c ( Sym ⁡ E ˇ ) 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 O P n ( − 1 ) 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 C, 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.