vector bundle, 2-vector bundle, (∞,1)-vector bundle
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higher geometry / derived geometry
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derived smooth geometry
Theorems
A theorem of Ehresmann 1951 states that a proper submersion of smooth manifolds is a locally trivial fibration (cf. Kolář, Slovák & Michor 1993, Lem. 92; Voisin 2002, Thm. 9.3).
The Ehresmann therem implies that the higher direct image of the constant sheaf on such are (-)local systems on .
(In the algebraic category, instead of the constant sheaf use the de Rham complex and instead of the higher direct images take the hyper-higher direct images.)
The corresponding vector bundle then has a canonical flat connection, known as the Gauss-Manin connection (e.g. Voisin 2002, Def. 9.13).
This is the typical setup one considers when studying variations of Hodge structure.
The theorem is named after:
Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable,
Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson et Cie., Paris (1951) 29–55
Séminaire Bourbaki, 1 24 (1952) 153–168 [numdam:SB_1948-1951__1__153_0, pdf]
Discussion in the context of natural operations on natural bundles:
Discussion in the context of Hodge theory:
See also
Last revised on February 1, 2024 at 11:06:33. See the history of this page for a list of all contributions to it.