nLab Ehresmann's theorem

Ehresmann’s theorem states that a proper submersion of smooth manifolds f:XYf : X \to Y is a locally trivial fibration.

This is important in algebraic geometry because it implies that the higher direct images R if *̲R^i f_\ast \underline{\mathbb{C}} of the constant sheaf ̲\underline{\mathbb{C}} on XX are (\mathbb{C}-)local systems on YY. (If we work in the algebraic category, then instead of the constant sheaf ̲\underline{\mathbb{C}} we take the de Rham complex Ω X \Omega_X^\bullet and instead of the higher direct images we take the hyper-higher direct images.) The corresponding vector bundle then has a canonical flat connection, known as the Gauss-Manin connection. This is the typical setup one considers when studying variations of Hodge structure.

Last revised on November 18, 2019 at 19:30:57. See the history of this page for a list of all contributions to it.