Integrality of the Betti moduli space

Johan de Jong, Columbia University
IAS - Simonyi Hall Seminar Room SH-101

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.