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

This is important in algebraic geometry because it implies that the higher direct images $R^i f_\ast \underline{\mathbb{C}}$ of the constant sheaf $\underline{\mathbb{C}}$ on $X$ are ($\mathbb{C}$-)local systems on $Y$. (If we work in the algebraic category, then instead of the constant sheaf $\underline{\mathbb{C}}$ we take the de Rham complex $\Omega_X^\bullet$ 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.