Some remarks on the companions conjecture for normal varieties

TL;DR

This paper explores the obstacles to extending the companions conjecture from smooth to normal varieties over finite fields, introducing a new property of morphisms and verifying it in specific cases.

Contribution

It introduces a new property of morphisms and demonstrates its applicability to prove the companions conjecture for certain singular normal varieties.

Findings

Verified the new property for some cases of normal varieties

Extended the companions conjecture to certain singular normal varieties

Identified obstructions to the conjecture for arbitrary normal 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.