nLab Ehresmann's theorem








A theorem of Ehresmann 1951 states that a proper submersion of smooth manifolds f:XYf : X \to Y is a locally trivial fibration (cf. Kolář, Slovák & Michor 1993, Lem. 92; Voisin 2002, Thm. 9.3).

Relation to Gauss-Manin connections

The Ehresmann therem implies that the higher direct image R if *̲R^i f_\ast \underline{\mathbb{C}} of the constant sheaf ̲\underline{\mathbb{C}} on such XX are (\mathbb{C}-)local systems on YY.

(In the algebraic category, instead of the constant sheaf ̲\underline{\mathbb{C}} use the de Rham complex Ω X \Omega_X^\bullet 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:

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.