Some remarks on the companions conjecture for normal varieties

TL;DR

The paper investigates extending the companions conjecture from smooth to normal varieties by introducing and studying $\lambda$-uniform morphisms. It defines $\lambda$-uniformity for morphisms and analyzes its behavior via étale fundamental groups and monodromy, proving key cases (normal $Z_0$, tree dual graphs in semi-stable curves, and open monodromy subgroups) where $\lambda$-uniformity holds and hence the conjecture transfers to contractions of suitable subvarieties. The results include a homotopy-invariance property of $\lambda$-uniformity and a framework to deduce the conjecture for certain singular contractions inside smooth ambient varieties. The work also discusses limitations via explicit examples (e.g., de Jong’s setup) and outlines directions such as pseudo-$\lambda$-uniformity to broaden applicability, signaling a path toward broader applicability of the companions conjecture to singular varieties.

Abstract

Drinfeld in 2010 proved the companions conjecture for smooth varieties over a finite field, generalizing L. Lafforgue's result for smooth curves. We study the obstruction to prove the conjecture for arbitrary normal varieties. To do this, we introduce a new property of morphisms. We verify this property in some cases, showing thereby the companions conjecture for some singular normal varieties.

Theorems & Definitions (26)

• Conjecture 1.1.1: Companions conjecture
• Lemma 1.2.1: Corollaire 6.11, SGA I, Exposé XI
• Theorem 1.2.2
• Conjecture 1.3.1: Conjecture \ref{['unif:c']}
• Theorem 1.3.2: Theorem \ref{['normal-uniform:t']}, Theorem \ref{['tree:t']}, and Theorem \ref{['finite:t']}
• Theorem 1.3.3
• Definition 2.1.1
• Theorem 2.1.2
• proof
• Definition 2.1.3