the linear differential equation .If extends to a one-form on the above will also define a local system on , so will be trivial since . So to give an interesting example, choose one with a pole at 0: in which case for , An n-sheeted covering map is a local system with fibers given by the set . Similarly, a fibre bundle with discrete fibre is a local system, because each path lifts uniquely to a given lift of its basepoint. (The definition adjusts to include set-valued local systems in the obvious way). A local system of k-vector spaces on X is equivalent to a k-linear representation of . If X is a variety, local systems are the same thing as D-modules which are additionally coherent O_X-modules (see O modules). If the connection is not flat (i.e. its curvature is nonzero), then parallel transport of a fibre F_x over x around a contractible loop based at x_0 may give a nontrivial automorphism of F_x, so locally constant sheaves can not necessarily be defined for non-flat connections. The Gauss–Manin connection is a prominent example of a connection whose horizontal sections are studied in relation to variation of Hodge structures. Generalization Local systems have a mild generalization to constructible sheaves -- a constructible sheaf on a locally path connected topological space is a sheaf such that there exists a stratification of where is a local system. These are typically found by taking the cohomology of the derived pushforward for some continuous map . For example, if we look at the complex points of the morphism then the fibers over are the smooth plane curve given by , but the fibers over are . If we take the derived pushforward then we get a constructible sheaf. Over we have the local systems while over we have the local systems where is the genus of the plane curve (which is ). Applications The cohomology with local coefficients in the module corresponding to the orientation covering can be used to formulate Poincaré duality for non-orientable manifolds: see Twisted Poincaré duality. See also Leray spectral sequence Gauss–Manin connection D-module Intersection homology Perverse sheaf References External links Discusses computing the cohomology with coefficients in a local system by using the twisted de Rham complex. Sheaf theory Algebraic topology