The Grothendieck Gamma Filtration, the Tits Algebras, and the J-invariant of a Linear Algebraic Group

Abstract

Consider a semisimple linear algebraic group G over an arbitrary field F, and a projective homogeneous G-variety X. The geometry of such varieties has been a consistently active subject of research in algebraic geometry for decades, with significant contributions made by Grothendieck, Demazure, Tits, Panin, and Merkurjev, among others. An effective tool for the classification of these varieties is the notion of a cohomological (or alternatively, a motivic) invariant. Two such invariants are the set of Tits algebras of G defined by J. Tits, and the J-invariant of G defined by Petrov, Semenov, and Zainoulline. Quéguiner-Mathieu, Semenov and Zainoulline discovered a connection between these invariants, which they developed through use of the second Chern class map. The first goal of the present thesis is to extend this connection through the use of higher Chern class maps. Our main technical tool is the Steinberg basis, which provides explicit generators for the γ-filtration on the Grothendieck group K_0(X) in terms of characteristic classes of line bundles over X. As an application, we establish a connection between the J-invariant and the Tits algebras of a group G of inner type E6. The second goal of this thesis is to relate the indices of the Tits algebras of G to nontrivial torsion elements in the γ-filtration on K_0(X). While the Steinberg basis provides an explicit set of generators of the γ-filtration, the relations are not easily computed. A tool introduced by Zainoulline called the twisted γ-filtration acts as a surjective image of the γ-filtration, with explicit sets of both generators and relations. We use this tool to construct torsion elements in the degree 2 component of the γ-filtration for groups of inner type D2n. Such a group corresponds to an algebra A endowed with an orthogonal involution having trivial discriminant. In the trialitarian case (i.e. type D4), we construct a specific element in the γ-filtration which detects splitting of the associated Tits algebras. We then relate the non-triviality of this element to other properties of the trialitarian triple such as decomposability and hyperbolicity.