Integrality of the Betti Moduli Space

Let X be a smooth projective variety over the complex numbers. Let M be the moduli space of irreducible representations of the topological fundamental group of X of a fixed rank r. Then M is a finite type scheme over the spectrum of the integers Z. One may ask if the irreducible components of M surject onto Spec(Z) or whether M is pure over Z in some sense. We give a weak answer to these questions and we discuss what other phenomena can be studied using the method of proof.



Johan De Jong


Columbia University