Tautological bundle

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In mathematics, tautological bundle is a term for a particularly natural vector bundle occurring over a Grassmannian, and more specially over projective space. Canonical bundle as a name 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.

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: see tautological line bundle. 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.

[edit] Properties

The Picard group of line bundles on $\mathbb P(V)$ is infinite cyclic, and the tautological line bundle is a generator.
In the case of projective space, where the tautological bundle is a line bundle, the associated invertible sheaf of sections is the tensor inverse of the Serre twist sheaf $\mathcal O(1)$ ; in other words the Serre sheaf is the other generator.