  The absolute sets of local systems on a smooth complex algebraic variety are
the subject of a conjecture of N. Budur and B. Wang based on an analogy with
special subvarieties of Shimura varieties. An absolute set should be the
higher-dimensional generalization of a local system of geometric origin. We
show that the conjecture for absolute sets of simple cohomologically rigid
local systems reduces to the zero-dimensional case, that is, to Simpson's
conjecture that every such local system with quasi-unipotent monodromy at
infinity and determinant is of geometric origin. In particular, the conjecture
holds for this type of absolute sets if the variety is a curve or if the rank
is two.
