  We show that closures of families of unitary local systems on quasiprojective
varieties for which the dimension of a graded component of Hodge filtration has
a constant value can be identified with a finite union of polytopes. We also
present a local version of the theorem. This yields the "Hodge decomposition"
of the set of unitary local systems with a non-vanishing cohomology extending
Hodge decomposition of characteristic varieties of links of plane curves
studied by the author earlier. We consider a twisted version of the
characteristic varieties generalizing the twisted Alexander polynomials.
Several explicit calculations for complements to arrangements are made.
