  Two decades ago, as part of their work of generic vanishing theorems,
Green-Lazarsfeld showed that over a compact Kahler manifold $X$, the cohomology
jump loci in the $Pic^\tau(X)$ are all translates of subtori. In this paper, we
generalize this result to higher dimensional vector bundles. It is showed by
Nadel that locally the moduli space of vector bundles with vanishing chern
classes is canonically isomorphic to a quadratic cone in the Zariski tangent
space of a point. We prove that under the isomorphism, the cohomology jump loci
are defined by linear equations.
