Examples of topological spaces with arbitrary cohomology jump loci

Abstract

Given any subvariety of a complex torus defined over $\mathbb{Z}$ and any positive integer $k$, we construct a finite CW complex $X$ such that the $k$-th cohomology jump locus of $X$ is equal to the chosen subvariety, and the $i$-th cohomology jump loci of $X$ are trivial for $i<k$.

Carlos Simpson has kindly pointed out to us that the main idea of this paper has been considered in his 1997 survey paper, at least for the case when $k=2$. This is added to the introduction