Non-abelian Rees construction and pure motives

TL;DR

This paper extends the classical Rees construction to a non-abelian setting using reductive groups, establishing a Galois correspondence and applying it to motives and algebraic cycles, with implications for algebraic geometry.

Contribution

It introduces a non-abelian Rees construction based on a Galois correspondence, generalizing classical results and providing new insights into motives and algebraic cycles.

Findings

New proof and generalization of Clozel-Deligne theorem

Galois correspondence between quasi-homogeneous spaces and monoidal categories

Applications to algebraic cycles and motives

Abstract

The classical Rees construction (of common use in commutative algebra and Hodge theory) interpolates between filtrations, viewed as ${\mathbb G}_m$-equivariant vector bundles on the affine line, and their associated gradings. Various non-abelian versions have been proposed, where the multiplicative group ${\mathbb G}_m$ is replaced by an arbitrary reductive group. Building on a construction due to P. O'Sullivan, we present a Galois correspondence between quasi-homogeneous spaces and certain monoidal categories, and apply it to monoidal categories of motives with concrete applications to algebraic cycles. In particular, we give a new proof and generalization of the Clozel-Deligne theorem about numerical equivalence on abelian varieties over finite fields.