nLab Ehresmann's theorem

Idea

A theorem of Ehresmann 1951 states that a proper submersion of smooth manifolds $f : X \to Y$ 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 $R^i f_\ast \underline{\mathbb{C}}$ of the constant sheaf $\underline{\mathbb{C}}$ on such $X$ are ($\mathbb{C}$-)local systems on $Y$.

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

Related concepts

• Gauss-Manin connection

References

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:

• Ivan Kolář, Jan Slovák, Peter Michor, Lemma 9.2 in: Natural operations in differential geometry, Springer (1993) [book webpage, doi:10.1007/978-3-662-02950-3, pdf]

Discussion in the context of Hodge theory:

• Claire Voisin (translated by Leila Schneps), Thm. 9.3 in Hodge theory and Complex algebraic geometry I, Cambridge Stud. in Adv. Math. 76 (2002) [doi:10.1017/CBO9780511615344, Gauss-Manin connection def: pdf]

See also

• Wikipedia, Ehresmann’s lemma