Moduli of $\ell$-adic pro-\'etale local systems for smooth non-proper schemes

Abstract

Let $X$ be a smooth scheme over an algebraically closed field. When $X$ is proper, it was proved in \cite{me1} that the moduli of $\ell$-adic continuous representations of $\pi_1^\et(X)$, $\LocSys(X)$, is representable by a (derived) $\Ql$-analytic space. However, in the non-proper case one cannot expect that the results of \cite{me1} hold mutatis mutandis. Instead, assuming $\ell$ is invertible in $X$, one has to bound the ramification at infinity of those considered continuous representations. The main goal of the current text is to give a proof of such representability statements in the open case. We also extend the representability results of \cite{me1}. More specifically, assuming $X$ is assumed to be proper, we show that $\LocSys(X)$ admits a canonical shifted symplectic form and we give some applications of such existence result.