flat vector bundle

(see also *Chern-Weil theory*, parameterized homotopy theory)

A *flat vector bundle* is a vector bundle with connection on a vector bundle which is flat. Specifically a *flat line bundle* is a line bundle with flat connection.

Under the identification of local systems – in the sense of locally constant and locally free abelian sheaves – with flat vector bundles, the flat connection induced on these vector bundles is called the *Gauss-Manin connection* (e.g. Voisin 2002, Def. 9.13).

Textbook accounts:

- Claire Voisin (translated by Leila Schneps), Section I 9.2.1 of:
*Hodge theory and Complex algebraic geometry I,*, Cambridge Stud. in Adv. Math.**76, 77**, 2002/3 (doi:10.1017/CBO9780511615344)

Last revised on July 25, 2021 at 08:35:38. See the history of this page for a list of all contributions to it.