Tautological bundle

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In mathematics, the tautological bundle is a particular natural vector bundle occurring over a Grassmannian: each Grassmannian has a single tautological bundle. In the case of projective space this is known as the tautological 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.

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 Vg 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 Vg 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 Vg, 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.

Properties[edit]

  • 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 \mathcal{O}(-1), the tensor inverse (ie the dual vector bundle) of the hyperplane bundle or Serre twist sheaf \mathcal{O}(1); in other words the hyperplane bundle is the generator of the Picard group having positive degree (as a divisor) and the tautological bundle is its opposite: the generator of negative degree.

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