Cohomology jump loci are generalizations of the usual cohomology groups of a topological space. In the first part of the talk, I will give a survey on the structure theorems of the cohomology jump loci. In the second part of the talk, I will construct (real) 6-dimensional compact symplectic-complex Calabi-Yau manifolds to illustrate how cohomology jump loci can give new constraints on the topology of compact Kahler manifolds.