Lectures on local systems #1

Lectures by Hélène Esnault

Abstract: In the last years, I studied arithmetic properties of local systems, be they complex or $\ell$-adic, be they in complex geometry or in geometry over finite fields. Why local systems and not fundamental groups directly? This is because we know extremely little on fundamental groups. Over $\mathbb{C}$, we know that a smooth quasi-projective variety has the homotopy type of a finite $CW$-complex, so its topological  fundamental group is finitely presented, but we essentially know nothing on how among finitely presented groups the fundamental groups coming from algebraic geometry are distinguished. However, we can hope to see more on the linear representations of the fundamental group, complex linear in topology or continuous $\ell$-adic arithmetically.

The first two sessions shall be devoted to proving similarities and differences between the topological fundamental group $\pi_1(X(\mathbb{C}),x)$ over $\mathbb{C}$  and the étale fundamental group $\pi_1(X_k,x)$ over an algebraically closed characteristic $p>0$ field $k$:

1) Similarity: if $X$ is smooth and admits a good compactification, then the tame quotient $\pi^t_1(X_k, x)$ of $\pi_1(X_k, x)$ (for example if $X$ is already projective smooth, then $\pi_1(X_k,x)$ itself) is finitely presented in the pro-finite sense.

2) Difference: assume $X$ projective. There is a property of $\pi_1(X_k,x)$, which we defined, which is fulfilled if $X$ is the mod $p$ reduction of a complex variety, but which is not always fulfilled.

1) was proven with Mark Shusterman and V. Srinivas and 2) with V. Srinivas and Jakob Stix, in both cases during the pandemic. On 2) I will also recall some of the classical obstructions to liftability and show how this new one differs.

In the Lectures 3-5, I will talk on integrality properties of the Betti moduli spaces of smooth complex quasi-projective varieties, the complex points of which parametrize complex semi-simple local systems. This is joint work disseminated in articles with Moritz Kerz, Michael Groechenig and Johan de Jong. We use here heavy arithmetic methods (Langlands program, both arithmetic and geometric). This produces a new obstruction for a finitely presented group to be the fundamental group of a smooth complex quasi-projective variety, which seems to be of a new kind (joint with Johan de Jong).