Non-abelian Rees construction and pure motives
Yves André
TL;DR
The paper develops a non-abelian analogue of the Rees construction by replacing the classical G_m-interpolation with a reductive group G, establishing a Galois-type anti-equivalence between even tensor categories and fix-pointed affine G-schemes X. Central to the framework is O'Sullivan’s construction, which associates to any even tensor category C a deformation space X and a pair of compatible tensor functors linking C to its semisimple quotient bar{C}, enabling a precise description of how non-abelian filtrations deform to gradings. The restricted construction refines this by imposing finiteness and tannakian-hull conditions, yielding a more computable quasi-homogeneous setting Vec_G(X) that captures many motivating examples, including multifiltrations and the Fibonacci category. Applications to pure motives show how X encodes obstructions to Grothendieck’s standard conjectures: when Künneth projectors are algebraic and signs hold, X collapses to a point, thereby recovering semisimplicity results and providing a new perspective on numerical versus homological equivalence, including a generalized Clozel-Deligne theorem for abelian varieties over finite fields. Overall, the work provides a Galois-type dictionary between monoidal, motivic categories and quasi-homogeneous G-varieties, with concrete computational leverage for questions in motive theory and algebraic cycles.
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.