Tautological bundle

In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of ${\displaystyle k}$-dimensional subspaces of ${\displaystyle V}$, given a point in the Grassmannian corresponding to a ${\displaystyle k}$-dimensional vector subspace ${\displaystyle W\subseteq V}$, the fiber over ${\displaystyle W}$ is the subspace ${\displaystyle W}$ 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^[1]) 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

${\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1),}$

the dual of the hyperplane bundle or Serre's twisting sheaf ${\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(1)}$. The hyperplane bundle is the line bundle corresponding to the hyperplane (divisor) ${\displaystyle \mathbb {P} ^{n-1}}$ in ${\displaystyle \mathbb {P} ^{n}}$. The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space.^[2]

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.