- Tautological bundle
In
mathematics , "tautological bundle" is a term for a particularly naturalvector bundle occurring over aGrassmannian , and more specially overprojective 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 thecanonical class inalgebraic geometry could scarcely be avoided.Grassmannians by definition are the parameter spaces for
linear subspace s, of a given dimension, in a givenvector 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 thedisjoint 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 thedual 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 functional s on "V"*. If "V" has dimension "n" + 1, the tautologicalline bundle is one tautological bundle, and the other, just described, is of rank "n".Properties
* The
Picard group of line bundles on isinfinite cyclic , and thetautological line bundle is a generator.
* In the case of projective space, where the tautological bundle is a line bundle, the associatedinvertible sheaf of sections is the tensor inverse of the Serre twist sheaf ; in other words the Serre sheaf is the "other" generator.ee also
*
Hopf bundle
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