From Flag Manifolds to Severi-Brauer Varieties: Intersection Theory, Algebraic Cycles and Motives

Abstract

The study of algebraic varieties originates from the study of smooth manifolds. One of the focal points is the theory of differential forms and de Rham cohomology. It’s algebraic counterparts are given by algebraic cycles and Chow groups. Linearizing and taking the pseudo-abelian envelope of the category of smooth projective varieties, one obtains the category of pure motives. In this thesis, we concentrate on studying the pure Chow motives of Severi-Brauer varieties. This has been a subject of intensive investigation for the past twenty years, with major contributions done by Karpenko, [Kar1], [Kar2], [Kar3], [Kar4]; Panin, [Pan1], [Pan2]; Brosnan, [Bro1], [Bro2]; Chernousov, Merkurjev, [Che1], [Che2]; Petrov, Semenov, Zainoulline, [Pet]; Calmès, [Cal]; Nikolenko, [Nik]; Nenashev, [Nen]; Smirnov, [Smi]; Auel, [Aue]; Krashen, [Kra]; and others. The main theorem of the thesis, presented in sections 4.3 and 4.4, extends the result of Zainoulline et al. in the paper [Cal] by providing new examples of motivic decompositions of generalized Severi-Brauer varieties.