Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-17T20:39:18Z No. of bitstreams: 1 alberthjohnnunezsullca.pdf: 789070 bytes, checksum: 6fff839362c420dcaaaf67f1f9975a5e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-04-18T13:51:59Z (GMT) No. of bitstreams: 1 alberthjohnnunezsullca.pdf: 789070 bytes, checksum: 6fff839362c420dcaaaf67f1f9975a5e (MD5) / Made available in DSpace on 2017-04-18T13:52:00Z (GMT). No. of bitstreams: 1 alberthjohnnunezsullca.pdf: 789070 bytes, checksum: 6fff839362c420dcaaaf67f1f9975a5e (MD5) Previous issue date: 2017-03-17 / Apresentamos neste trabalho uma decomposição celular CW das variedades Grassmannianas via teoria de Morse. Isto é feito de duas maneiras distintas por meio de representações matriciais das Grassmannianas chamadas modelo projeção e modelo reflexão. Definimos funções de Morse, a saber, uma função do tipo altura e uma função do tipo “distância ao quadrado”, respectivamente, para cada um dos modelos projeção e reflexão. Estudamos os seus pontos críticos e os índices dos mesmos, obtendo assim duas formas para calcular a decomposição celular CW. Em particular, no modelo projeção, isto é feito exibindo-se as curvas integrais associadas ao campo gradiente da função altura. / We present in this work a CW cellular decomposition of Grassmannian varieties via Morse theory. This is done in two different ways. By means of matrix representations of Grassmannian called model projection and reflection model. We define Morse functions, namely a height-type function and a "square-distance" function, respectively, for each of the projection and reflection models. We study their critical points and their indices, thus obtaining two ways to calculate the CW cellular decomposition. In particular, in the projection model, this is done by displaying the integral curves associated with the gradient field of the height function.

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.