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.
This video is part of the Institute for Advanced Study‘s Arithmetic geometry seminar.
