  We apply the method of Dimca-Papadima to study the cohomology jump loci in
the representation variety and the moduli space of vector bundles with
vanishing chern classes for a compact K\"ahler manifold. We introduce modules
over differential graded Lie algebra to extend results of Dimca-Papadima to a
general point in the representation variety or the moduli space. We show that
locally the cohomology jump loci is isomorphic to the resonance variety via the
exponential map. This paper generalizes a previous result of the author.
