Orateur
Description
This is a report on joint work with Hélène Esnault. Let $X$ be a smooth projective variety over the complex numbers $\mathbb{C}$. 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 $\mathbb{Z}$. We may ask whether $M$ is pure over $\mathbb{Z}$ in the sense of Raynaud-Gruson, for example we can ask if the irreducible components of $M$ which dominate ${\rm Spec}(\mathbb{Z})$ actually surject onto ${\rm Spec}(\mathbb{Z})$. We will explain what this means, present a weak answer to this question, apply this to exclude some abstract groups as the fundamental groups of smooth projective varieties over $\mathbb{C}$, and we discuss what other phenomena can be studied using the method of proof.
