Affine representations of the fundamental group (Q2716424)

There is a classical correspondence between some unitary representations of the fundamental group of a compact Kähler manifold and polystable vector bundles. This work considers representations of these fundamental groups into the \(n\)-dimensional complex affine group. The main result describes the objects associated to it, in the vector bundle framework. An affine representation as above gives a linear part and a translation part. The paper describes how the linear part (when it is semisimple) gives a polystable Higgs bundle and the translation part corresponds to an element of a generalized de Rham cohomology (on the bundles). Also, the same problem for representations into a parabolic subgroup of the \(n\)-dimensional general linear group is studied.