cohomology jump loci are generalizations of usual cohomology groups of a topological space. in the first part of the talk, i will give a survey on the recent development of the theory of cohomology jump loci of complex algebraic varieties. in the second part of the talk, i will use some concrete examples of (real) 6-dimensional symplectic-complex calabi-yau manifolds to illustrate how cohomology jump loci can give new constraints on the topology of compact kahler manifolds.