Moduli of ℓ-adic pro-étale local systems for smooth non-proper schemes

2019/04/16 by António, Jorge
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.08001

Abstract

Let X be a smooth scheme over an algebraically closed field. When X is proper, it was proved in \citeme1 that the moduli of ℓ-adic continuous representations of π₁^\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 \citeme1 hold mutatis mutandis. Instead, assuming ℓ 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 \citeme1. 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.

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