Tautological bundle on a Grassmannian - Imported Knowlpedia Content

Core idea

Let $V$ be a finite-dimensional real or complex vector space and let $\operatorname{Gr}_k(V)$ be its Grassmannian. The tautological bundle is

\mathcal S=\{(W,v)\in\operatorname{Gr}_k(V)\times V:v\in W\}, \qquad \pi(W,v)=W.

Its fiber $\mathcal S_W$ over a $k$ -plane $W$ is canonically $W$ itself. The graph charts on the Grassmannian give $\mathcal S$ the structure of a rank- $k$ vector bundle and identify it as a vector subbundle of the trivial bundle $\operatorname{Gr}_k(V)\times V$ . The corresponding quotient bundle $\mathcal Q=(\operatorname{Gr}_k(V)\times V)/\mathcal S$ has fiber $\mathcal Q_W\cong V/W$ .

Universal property

Let $E\subseteq X\times V$ be a rank- $k$ vector subbundle of a trivial bundle over a smooth manifold $X$ . Sending $x$ to the subspace $E_x\subseteq V$ defines a smooth classifying map $c:X\to\operatorname{Gr}_k(V)$ , and there is a canonical bundle isomorphism

E\cong c^*\mathcal S.

Conversely, every pullback of $\mathcal S$ is such a subbundle. This finite-dimensional universal property classifies subbundles equipped with an embedding into a fixed trivial bundle; see Husemoller, chapters on Grassmannians and universal bundles.

Exact sequence and characteristic classes

The inclusion and quotient fit into the canonical short exact sequence

0\longrightarrow\mathcal S\longrightarrow \operatorname{Gr}_k(V)\times V\longrightarrow\mathcal Q\longrightarrow0.

The characteristic classes of $\mathcal S$ and $\mathcal Q$ generate much of the cohomology used in Grassmannian and Schubert calculus. Their relation follows from the triviality of the middle bundle; see Milnor–Stasheff, §5.

Standard examples and conventions

On $\operatorname{Gr}_1(\mathbb R^n)=\mathbb{RP}^{n-1}$ , $\mathcal S$ is the real tautological line bundle. On $\operatorname{Gr}_1(\mathbb C^n)=\mathbb{CP}^{n-1}$ , it is the complex tautological line bundle, commonly denoted $\mathcal O(-1)$ . The word “canonical bundle” should not be used here without qualification, because in complex geometry it usually denotes the top exterior power of the holomorphic cotangent bundle.

References

John W. Milnor and James D. Stasheff, Characteristic Classes, Princeton University Press, 1974. DOI record. Relevant: §5, Grassmannians, universal bundles, and complementary bundles.
Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: chapters on Grassmannians and universal vector bundles.