Higgs line bundles, Green-Lazarsfeld sets, and maps of Kähler manifolds to curves

Let X be a compact Kähler manifold. The set $\operatorname{char}(X)$ of one-dimensional complex valued characters of the fundamental group of X forms an algebraic group. Consider the subset of $\operatorname{char}(X)$ consisting of those characters for which the corresponding local system has nontrivial cohomology in a given degree d. This set is shown to be a union of finitely many components that are translates of algebraic subgroups of $\operatorname{char}(X)$. When the degree d equals 1, it is shown that some of these components are pullbacks of the character varieties of curves under holomorphic maps. As a corollary, it is shown that the number of equivalence classes (under a natural equivalence relation) of holomorphic maps, with connected fibers, of X onto smooth curves of a fixed genus $> 1$ is a topological invariant of X. In fact it depends only on the fundamental group of X.