Ehresmann’s theorem states that a proper submersion of smooth manifolds is a locally trivial fibration.
This is important in algebraic geometry because it implies that the higher direct images of the constant sheaf on are (-)local systems on . (If we work in the algebraic category, then instead of the constant sheaf we take the de Rham complex 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.