Tautological bundle

In the mathematical fields of topology and geometry , the tautological bundle on a projective space is an object that assigns the straight line from which it was created to each point.

definition

The tautological bundle over a projective space to a vector space is the straight line bundle whose fiber at a point is the corresponding one-dimensional subspace of . It is a sub-bundle of the trivial bundle . ${\displaystyle \mathbb {P} (V)}$ ${\displaystyle V}$ ${\displaystyle x\in \mathbb {P} (V)}$ ${\displaystyle x}$ ${\displaystyle V}$ ${\displaystyle \mathbb {P} (V)\times V}$

Analogously, the tautological bundle can be defined on the Graßmannian of the -dimensional subspaces of a vector space; it is a vector bundle of rank . ${\displaystyle k}$ ${\displaystyle k}$

properties

• The Picard group of straight line bundles on is infinitely cyclic , and the tautological bundle is a generator. ${\displaystyle \mathbb {P} (V)}$
• The sheaf of cuts in the tautological bundle is inverse to Serre's twisting sheaf . ${\displaystyle {\mathcal {O}}(1)}$

See also

• Topological K-theory

literature

• Phillip Griffiths , Joe Harris : Principles of algebraic geometry. Wiley, New York NY et al. 1994, ISBN 0-471-05059-8 .
• Robin Hartshorne : Algebraic Geometry (= Graduate Texts in Mathematics 52). Springer-Verlag, Berlin et al. 1977, ISBN 0-387-90244-9 .